Weyl Sets in a Truncated Matricial Stieltjes Moment Problem

Abstract

Given a point w in the open upper complex half-plane $\Pi _{\mathord {+}}$, we describe the set of possible values $F_\sigma \left( {w}\right)$ of Stieltjes transforms $F_\sigma \left( {z}\right) :=\int _{[\alpha ,\infty )}\left( {x-z}\right) ^{-1}\sigma \left( {\textrm{d}x}\right)$, $z\in \Pi _{\mathord {+}}$, corresponding to solutions $\sigma$ to a truncated matricial Stieltjes moment problem as intersection of two matrix balls.

1 Introduction

This paper covers a research issue which arises from the following truncated matricial Stieltjes type moment problem: Given a finite sequence $(s_j)_{j=0}^{m}$ of complex ${q\times q}$ matrices, describe the set of all non-negative Hermitian ${q\times q}$ measures $\sigma$ which satisfy

$$\begin{aligned} s_{j} =\int _{[\alpha ,\infty )}t^j\sigma \left( {\textrm{d}t}\right) \end{aligned}$$

for every choice of $j\in \left\{ {0,\dotsc ,m-1} \right\}$ as well as

$$\begin{aligned} s_{m} \succcurlyeq \int _{[\alpha ,\infty )}t^m\sigma \left( {\textrm{d}t}\right) . \end{aligned}$$

In fact, the solutions $\sigma$ to this matricial moment problem are in one-to-one correspondence with certain holomorphic matrix functions F. The core objective of our investigations is to characterize the set of all possible values $F\left( {w}\right)$ which these matrix functions can take at a fixed point w of the open upper complex half-plane $\Pi _{\mathord {+}}$. (The instance of this problem for $w\in (-\infty ,\alpha )$ was treated in [13, Sec. 17].) In our approach, we will apply an idea which has previously been employed by Krein and Nudelman in [20] (see in particular, [20, Ch. 5]) to solve the univariate case of this problem, who themselves refer to methods from [18]. There, Henrici and Pflüger investigate special sets of values in the context of estimates for Stieltjes fractions. Seen that solutions to a Stieltjes moment problem can be immediately adapted to solve a corresponding Hamburger moment problem, it is obvious that the values of the Stieltjes transforms of the solutions to the Stieltjes moment problem at a given point w lie within the set of values of the Stieltjes transforms of the solutions to the Hamburger moment problem at this same point. Building on this, the formulation of our approach indeed lies in reducing the Stieltjes moment problem at hand to two interrelated moment problems of Hamburger type. The first one emerges naturally as mentioned above. The second Hamburger moment problem belongs to a sequence $(a_j)_{j=0}^{m-1}$ of modified data, which incorporates the left interval boundary $\alpha$ of the integral domain ${[\alpha ,\infty )}$. As proved in [6], the set of values $F\left( {w}\right)$ of the $\mathbb {R}$-Stieltjes transforms F of the solutions $\sigma$ to a considered Hamburger moment problem at a fixed point w coincides with some matrix ball, the center and left and right semi-radii of which can be explicitly expressed in terms of the prescribed data. Assigning the respective matrix ball to each of the two Hamburger moment problems allocated to the Stieltjes moment problem under consideration, it turns out that the set in question is indeed a subset of the intersection of these two matrix balls. Even more, the values of all ${[\alpha ,\infty )}$-Stieltjes transforms of the solutions in the Stieltjes case at a single point $w\in \Pi _{\mathord {+}}$ actually fill in that intersection. Verifying this assertion is proved to be more difficult than the converse inclusion. As to be seen throughout this paper, various polynomial systems with orthogonality properties will play a central role within this proof. Both the Hamburger and the Stieltjes moment problems to a given sequence are each assigned such system of polynomials, which inter alia, appear in representations of the treated Stieltjes transforms as linear fractional transformations of pairs of certain matrix functions. Accordingly, results worked out in [13], where such representations are stated for Stieltjes moment problems, are one essential ingredient of our approach. Going into detail, the aim is to find an ${[\alpha ,\infty )}$-Stieltjes transform F of a solution $\sigma$ to the Stieltjes moment problem, such that its value $F\left( {w}\right)$ at the prescribed point w coincides with an arbitrarily given matrix X belonging to the intersection of the two matrix balls.

We describe the procedure in the case $m=2n$ with an arbitrarily given positive integer n. By inserting a certain constant Hamburger parameter pair $\left( {\phi };{\psi }\right)$ into a linear fractional transformation corresponding to the Hamburger moment problem associated to the given sequence $(s_j)_{j=0}^{2n}$, we first construct a rational $\mathbb {R}$-Stieltjes transform of a certain solution to this Hamburger problem, the value of which at the point w coincides with X. Analogously, inserting a certain constant Hamburger parameter pair $\left( {\phi _{\mathord {\circ }}};{\psi _{\mathord {\circ }}}\right)$ into a linear fractional transformation corresponding to the second Hamburger moment problem associated to the sequence $(a_j)_{j=0}^{2n-2}$, we get a rational $\mathbb {R}$-Stieltjes transform of a certain solution to this second Hamburger problem, the value of which at w coincides with $\left( {w-\alpha }\right) X+s_{0}$. Afterwards modifying the first constant Hamburger parameter pair $\left( {\phi };{\psi }\right)$ to a Stieltjes parameter pair $\left( {\phi _{\mathord {\bullet }}};{\psi _{\mathord {\bullet }}}\right)$ corresponding to the ${[\alpha ,\infty )}$-Stieltjes transform of a solution to the considered Stieltjes moment problem with value X at w, proves to be difficult and relies on close interrelations of the polynomial systems touched upon above that appear in parametrizations of the Stieltjes transforms of the solutions to the different moment problems at hand. However, this can be done by additionally applying the J-properties of the two Hamburger parameter pairs $\left( {\phi };{\psi }\right)$ and $\left( {\phi _{\mathord {\circ }}};{\psi _{\mathord {\circ }}}\right)$.

In each setting, i. e., whether we regard the Stieltjes or the two Hamburger moment problems, we use different yet related terminology. Recalling that Stieltjes transforms of solutions to either moment problem can be written as certain linear fractional transformations, switching from one case to the other one requires a conversion of both the coefficient functions and the parameter pairs appearing in these transformations. Concerning the latter, the parameters in the Hamburger case are so-called Nevanlinna pairs, which are pairs of matrix-valued functions meromorphic in $\Pi _{\mathord {+}}$ that fulfill some additional conditions. In contrast, the pairs used as parameters in the context of the Stieltjes moment problem consist of matrix-valued functions meromorphic in $\mathbb {C}\backslash {[\alpha ,\infty )}$, which satisfy conditions partly resembling those in the Hamburger case. To prove interrelations between the mentioned linear fractional transformations, we also use Nevanlinna pairs extended to $\mathbb {C}\backslash \mathbb {R}$ by a sort of reflection principle.

This paper is organized as follows. Section 2 contains some preliminaries and notations. In Sect. 3, we state some basic facts on the Herglotz–Nevanlinna class $\mathcal {R}_{q}\left( {\Pi _{\mathord {+}}}\right)$ of matrix-valued holomorphic functions in the open upper half-plane $\Pi _{\mathord {+}}$. Section 4 is dedicated to the discussion of two types of Nevanlinna pairs, namely in $\Pi _{\mathord {+}}$ and $\mathbb {C}\backslash \mathbb {R}$, respectively. These are classes of meromorphic matrix-valued functions, the latter of which is used to rewrite the linear fractional transformation corresponding to each of the two associated matricial Hamburger moment problems in terms of a single system of matrix polynomials adapted to the underlying Stieltjes moment problem. Section 5 is written against the background of matricial Hamburger moment problems. We recall the notion of the $\mathcal {H}$-parameter sequence $(\mathfrak {h}_{j})_{j=0}^{\kappa }$ which is associated with a sequence $(s_j)_{j=0}^{\kappa }$ of complex matrices (see Definition 5.2) and consider the $\mathbb {R}$-quadruple of matrix polynomials $\left[ {(\mathfrak {a}_{k})_{k=0}^{\dot{\kappa }},(\mathfrak {b}_{k})_{k=0}^{\dot{\kappa }},(\mathfrak {c}_{k})_{k=0}^{\dot{\kappa }},(\mathfrak {d}_{k})_{k=0}^{\dot{\kappa }}} \right]$ (abbreviating $\mathbb {R}$-QMP) associated with $(s_j)_{j=0}^{\kappa }$. In Sect. 6, we recall the Weyl matrix balls (see [6, Thm. 8.7]) which are associated with a truncated matricial Hamburger moment problems. Starting with Sect. 7, the previous preparations are now applied to the truncated matricial Stieltjes problem itself. For this reason, we recall the notion of ${[\alpha ,\infty )}$-non-negative definite (resp., ${[\alpha ,\infty )}$-non-negative definite extendable) sequences of matrices. Moreover, we introduce several classes of holomorphic matrix-valued functions in $\mathbb {C}\backslash {[\alpha ,\infty )}$ and discuss corresponding integral representations (see Theorem 7.7). In Sect. 8, we consider the class $\mathcal{P}\mathcal{S}_{q}\left( {\mathbb {C}\backslash {[\alpha ,\infty )}}\right)$ of pairs of meromorphic matrix-valued functions which were used in [13] as parameters for the parametrization of the solution set of the truncated matricial Stieltjes moment problem (see Definition 8.1). An essential aspect here is the construction of such pairs in Lemma 8.5 with prescribed value in a given point $\gamma \in \mathbb {C}\backslash \mathbb {R}$. In Sect. 9, we discuss some basic results on the $\mathcal {K}_\alpha$-parameter sequence $(\mathfrak {k}_{j})_{j=0}^{\kappa }$. In Sect. 10, we study some matrix polynomials $\textbf{p}_{\ell }$, $\textbf{q}_{\ell }$, $\textbf{r}_{\ell }$, and $\textbf{t}_{\ell }$ which are associated with the matricial Stieltjes moment problem. The construction of the matrix polynomials $\textbf{p}_{\ell }$ and $\textbf{q}_{\ell }$ originate in [13]. In Theorems 10.17 and 10.18, we recall the parametrization of the solution set of the truncated matricial Stieltjes moment problem which was obtained in [13, Theorems 15.6 and 15.7] in terms of these polynomials. Sections 11 and 12 are devoted to the study of the subsystems $\left[ {(\textbf{a}_{k})_{k=0}^{\dot{\kappa }},(\textbf{b}_{k})_{k=0}^{\dot{\kappa }},(\textbf{c}_{k})_{k=0}^{\dot{\kappa }},(\textbf{d}_{k})_{k=0}^{\dot{\kappa }}} \right]$ and $\left[ {(\textbf{a}_{{{\mathord {\circ }},k}})_{k=0}^{\ddot{\kappa }},(\textbf{b}_{{{\mathord {\circ }},k}})_{k=0}^{\ddot{\kappa }},(\textbf{c}_{{{\mathord {\circ }},k}})_{k=0}^{\ddot{\kappa }},(\textbf{d}_{{{\mathord {\circ }},k}})_{k=0}^{\ddot{\kappa }}} \right]$ of the matrix polynomials studied in Sect. 10. In [13, Propositions 14.7 and 14.8], it is shown that the systems $\left\{ {\textbf{b}_{k}} \right\}$ and $\left\{ {\textbf{b}_{{{\mathord {\circ }},k}}} \right\}$ are monic right orthogonal systems for the sequences $\left\{ {s_{j}} \right\}$ and $\left\{ {a_{j}} \right\}$ given by (7.1), respectively. Proposition 11.13 and Corollary 11.14 (resp., Proposition 12.16 and Corollary 12.17) contain results concerning linear fractional transformations synchronizing the interplay between Hamburger and Stieltjes matricial moment problems. In Sect. 13, we prove that the values of Stieltjes transforms of solutions of the truncated matricial Stieltjes moment problem belong to the intersection of the two matrix balls associated with the corresponding matricial Hamburger moment problems. Conversely, in Sects. 14 and 15, we show that each matrix X from the intersection of the two matrix balls occurs as value $F\left( {w}\right)$ of the Stieltjes transform F at w of a solution $\sigma$ of the Stieltjes moment problem under consideration. The case of a sequence $(s_j)_{j=0}^{2n}$ of prescribed moments with an arbitrary integer $n>0$ is dealt with in Sect. 14, while the case of a sequence $(s_j)_{j=0}^{2n+1}$ of prescribed moments with an arbitrary integer $n\ge 0$ is treated in Sect. 15. Section 16 is dedicated to the case that only the 0th moment $s_{0}$ is prescribed. The paper is supplemented by two Appendices A and B with special results on matrix theory and on the integration theory with respect to non-negative Hermitian measures, respectively.

2 Preliminaries and Notation

Let $\mathbb {C}$, $\mathbb {R}$, $\mathbb {N}_0$, and $\mathbb {N}$ be the set of all complex numbers, the set of all real numbers, the set of all non-negative integers, and the set of all positive integers, respectively. Further, for every choice of $\upsilon ,\omega \in \mathbb {R}\cup \left\{ {-\infty ,\infty } \right\}$, let $\mathbb {Z}_{\upsilon ,\omega }$ be the set of all integers k such that $\upsilon \le k\le \omega$. Throughout this paper, if not explicitly mentioned otherwise, let $p,q,r\in \mathbb {N}$.

If ${\mathcal {X}}$ is a non-empty set, then ${\mathcal {X}}^{p\times q}$ represents the set of all ${p\times q}$ matrices each entry of which belongs to ${\mathcal {X}}$, and ${\mathcal {X}}^p$ is short for ${\mathcal {X}}^{p\times 1}$. The notation $\mathbb {C}_\textrm{H}^{{q\times q}}$ is used to denote the set of all Hermitian complex ${q\times q}$ matrices. We write $\mathbb {C}_\succcurlyeq ^{{q\times q}}$ to designate the set of all non-negative Hermitian complex ${q\times q}$ matrices.

Let $\left( {\Omega ,\mathfrak {A}}\right)$ be a measurable space. Then each countably additive mapping defined on ${\mathfrak {A}}$ with values in $\mathbb {C}_\succcurlyeq ^{{q\times q}}$ is called a non-negative Hermitian ${q\times q}$ measure on $\left( {\Omega ,\mathfrak {A}}\right)$ and the notation ${\mathcal {M}_\succcurlyeq ^{q}(\Omega ,{\mathfrak {A}})}$ stands for the set of all non-negative Hermitian ${q\times q}$ measures on $\left( {\Omega ,\mathfrak {A}}\right)$. Let $\mu =\left[ {\mu _{jk}} \right] _{j,k=1}^{q}$ be a non-negative Hermitian ${q\times q}$ measure on $\left( {\Omega ,\mathfrak {A}}\right)$. Then we use $\mathcal {L}^{1}\left( {\Omega ,\mathfrak {A},\mu ;\mathbb {C}}\right)$ to denote the set of all Borel measurable functions $f:\Omega \rightarrow \mathbb {C}$ for which $\int _\Omega |{f} |\textrm{d}\nu _{jk}<\infty$ holds true for every choice of j and k in $\mathbb {Z}_{1,q}$, where $\nu _{jk}$ is the variation of the complex measure $\mu _{jk}$. If $f\in \mathcal {L}^{1}\left( {\Omega ,\mathfrak {A},\mu ;\mathbb {C}}\right)$, then let $\int _\Omega f\textrm{d}\mu :=\left[ {\int _\Omega f\textrm{d}\mu _{jk}} \right] _{j,k=1}^{q}$ and we also write $\int _\Omega f(\omega )\mu \left( {\textrm{d}\omega }\right)$ for this integral.

Denote by ${\mathfrak {B}_{\mathbb {R}}}$ (resp., ${\mathfrak {B}_{\mathbb {C}}}$) the $\sigma$-algebra of all Borel subsets of $\mathbb {R}$ (resp., $\mathbb {C}$). Let $\Omega \in {\mathfrak {B}_{\mathbb {R}}}\backslash \left\{ {\emptyset } \right\}$. Then designate by ${\mathfrak {B}_{\Omega }}$ the $\sigma$-algebra of all Borel subsets of $\Omega$ and by ${\mathcal {M}_\succcurlyeq ^{q}(\Omega )}$ the set of all non-negative Hermitian ${q\times q}$ measures on $\left( {\Omega ,{\mathfrak {B}_{\Omega }}}\right)$, i. e., ${\mathcal {M}_\succcurlyeq ^{q}(\Omega )}$ is short for ${\mathcal {M}_\succcurlyeq ^{q}(\Omega ,{\mathfrak {B}_{\Omega }})}$.

Throughout this paper, let $\kappa \in \mathbb {N}_0\cup \left\{ {\infty } \right\}$. We denote by $\mathcal {M}_{\succcurlyeq ,\kappa }^{q}(\Omega )$ the set of all $\sigma \in {\mathcal {M}_\succcurlyeq ^{q}(\Omega )}$ such that, for all $j\in \mathbb {Z}_{0, \kappa }$, the function $f_j:\Omega \rightarrow \mathbb {C}$ defined by $f_j(\omega ):=\omega ^j$ belongs to $\mathcal {L}^{1}\left( {\Omega ,{\mathfrak {B}_{\Omega }},\sigma ;\mathbb {C}}\right)$. If $\sigma \in \mathcal {M}_{\succcurlyeq ,\kappa }^{q}(\Omega )$, then, for all $j\in \mathbb {Z}_{0,\kappa }$, let $s_{j}^{\left( {\sigma }\right) }:=\int _\Omega \omega ^j\sigma \left( {\textrm{d}\omega }\right)$. For particular $\Omega \in {\mathfrak {B}_{\mathbb {R}}}\backslash \left\{ {\emptyset } \right\}$, the following moment problem is considered:

Problem

${\textsf{MP}[\Omega ;(s_j)_{j=0}^{m},\preccurlyeq ]}$ Let $m\in \mathbb {N}_0$ and let $(s_j)_{j=0}^{m}$ be a sequence of complex ${q\times q}$ matrices. Parametrize the set ${\mathcal {M}^{q}_\succcurlyeq [\Omega ;(s_j)_{j=0}^{m},\preccurlyeq ]}$ of all $\sigma \in \mathcal {M}_{\succcurlyeq ,m}^{q}(\Omega )$ for which the matrix $s_{m}-s_{m}^{\left( {\sigma }\right) }$ is non-negative Hermitian and, in the case $m\ge 1$, for which additionally $s_{j}=s_{j}^{\left( {\sigma }\right) }$ is fulfilled for all $j\in \mathbb {Z}_{0,m-1}$.

If $n\in \mathbb {N}_0$ and if $(s_j)_{j=0}^{2n}$ is a sequence of complex ${q\times q}$ matrices, then $(s_j)_{j=0}^{2n}$ is called $\mathbb {R}$-non-negative definite (or Hankel non-negative definite) if the block Hankel matrix

$$\begin{aligned} H_{n}:=\left[ {s_{j+k}} \right] _{j,k =0}^n \end{aligned}$$

is non-negative Hermitian. For all $n\in \mathbb {N}_0$, we will write $\mathcal {H}^\succcurlyeq _{q,2n}$ for the set of all sequences $(s_j)_{j=0}^{2n}$ of complex ${q\times q}$ matrices which are $\mathbb {R}$-non-negative definite. If $n\in \mathbb {N}_0$ and if $(s_j)_{j=0}^{2n}\in \mathcal {H}^\succcurlyeq _{q,2n}$, then, for each $m\in \mathbb {Z}_{0,n}$, the sequence $(s_j)_{j=0}^{2m}$ obviously belongs to $\mathcal {H}^\succcurlyeq _{q,2m}$. Thus, let $\mathcal {H}^\succcurlyeq _{q,\infty }$ be the set of all sequences $(s_j)_{j=0}^{\infty }$ of complex ${q\times q}$ matrices such that, for all $n\in \mathbb {N}_0$, the sequence $(s_j)_{j=0}^{2n}$ belongs to $\mathcal {H}^\succcurlyeq _{q,2n}$.

For all $n\in \mathbb {N}_0$, let $\mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,2n}$ be the set of all sequences $(s_j)_{j=0}^{2n}$ of complex ${q\times q}$ matrices for which there exist complex ${q\times q}$ matrices $s_{2n+1}$ and $s_{2n+2}$ such that $(s_j)_{j=0}^{2(n+1)}$ belongs to $\mathcal {H}^\succcurlyeq _{q,2\left( {n+1}\right) }$. Furthermore, for all $n\in \mathbb {N}_0$, we will use $\mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,2n+1}$ to denote the set of all sequences $(s_j)_{j=0}^{2n+1}$ of complex ${q\times q}$ matrices for which there exists a complex ${q\times q}$ matrix $s_{2n+2}$ such that $(s_j)_{j=0}^{2(n+1)}$ belongs to $\mathcal {H}^\succcurlyeq _{q,2\left( {n+1}\right) }$. For technical reasons, we set $\mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,\infty }:=\mathcal {H}^\succcurlyeq _{q,\infty }$. For each $\tau \in \mathbb {N}_0\cup \left\{ {\infty } \right\}$, the elements of the set $\mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,\tau }$ are called $\mathbb {R}$-non-negative definite extendable (or Hankel non-negative definite extendable) sequences.

We will write $I_{q}$ to denote the identity matrix in $\mathbb {C}^{{q\times q}}$, whereas $O_{{p\times q}}$ is the zero matrix belonging to $\mathbb {C}^{{p\times q}}$. Sometimes, if the size is clear from the context, we will omit the indices and write $I$ and $O$, respectively. For each $A\in \mathbb {C}^{{p\times q}}$, let $\mathcal {R}\left( {A}\right)$ be the column space of A, let $\mathcal {N}\left( {A}\right)$ be the null space of A, and let ${{\,\textrm{rank}\,}}A$ be the rank of A. For each $A\in \mathbb {C}^{{q\times q}}$, we will use $\Re A$ and $\Im A$ to denote the real part and the imaginary part of A, respectively: $\Re A:=\frac{1}{2}(A+A^*)$ and $\Im A:=\frac{1}{2\textrm{i}}(A-A^*)$. Furthermore, for each $A\in \mathbb {C}^{{p\times q}}$, let $\Vert {A} \Vert$ be the operator norm of A. A complex ${p\times q}$ matrix A is said to be contractive if $\Vert {A} \Vert \le 1$. We use $\mathbb {K}_{{p\times q}}$ in order to designate the set of all contractive complex ${p\times q}$ matrices. For each $A\in \mathbb {C}^{{p\times q}}$, let $A^{\mathord {+}}$ be the Moore–Penrose inverse of A. Given two complex matrices A and B, we use ${{\,\textrm{diag}\,}}\left( {A,B}\right)$ to denote the corresponding block diagonal matrix. If A and B are Hermitian complex ${q\times q}$ matrices, then we will write $A\preccurlyeq B$ (or $B\succcurlyeq A$) to indicate that $B-A$ is a non-negative Hermitian matrix.

For all $x,y\in \mathbb {C}^{q}$, by $\langle {x},{y}\rangle _\textrm{E}$ we denote the (left-hand side) Euclidean inner product of x and y, i. e., we have $\langle {x},{y}\rangle _\textrm{E}:=y^*x$. If $\mathcal {M}$ is a non-empty subset of $\mathbb {C}^{q}$, then let $\mathcal {M}^\bot$ be the set of all vectors in $\mathbb {C}^{q}$ which are orthogonal to $\mathcal {M}$ (with respect to the Euclidean inner product $\langle {.},{.}\rangle _\textrm{E}$). If $\mathcal {U}$ is a linear subspace of $\mathbb {C}^{q}$, then let $\mathbb {P}_{\mathcal {U}}$ be the orthogonal projection matrix onto $\mathcal {U}$.

If ${\mathcal {W}}$, ${\mathcal {X}}$, and ${\mathcal {Y}}$ are non-empty sets with ${\mathcal {W}}\subseteq {\mathcal {X}}$ and if $\varphi :{\mathcal {X}}\rightarrow {\mathcal {Y}}$ is a mapping, then ${{\,\textrm{Rstr}\,}}_{\mathcal {W}}\varphi$ marks the restriction of $\varphi$ onto ${\mathcal {W}}$. Let $\mathcal {G}$ be a non-empty subset of $\mathbb {C}$. Then let

$$\begin{aligned} \mathcal {G}^{\mathord {\vee }}:=\left\{ {\overline{z}}:{z\in \mathcal {G}}\right\} . \end{aligned}$$

(2.1)

If $f:\mathcal {G}\rightarrow \mathbb {C}$ is a complex-valued function, then let $\mathcal {Z}\left( {f}\right) :=\left\{ {z\in \mathcal {G}}:{f\left( {z}\right) =0}\right\}$. If $F:\mathcal {G}\rightarrow \mathbb {C}^{{p\times q}}$ is a matrix-valued function, then let $F^{\mathord {\vee }}:\mathcal {G}^{\mathord {\vee }}\rightarrow \mathbb {C}^{{q\times p}}$ be defined by

$$\begin{aligned} F^{\mathord {\vee }}\left( {z}\right) :=\left[ {F\left( {\overline{z}}\right) } \right] ^*. \end{aligned}$$

(2.2)

Now let $\mathcal {G}$ be a non-empty open subset of $\mathbb {C}$. Then we will call a subset $\mathcal {D}$ of $\mathcal {G}$ a discrete subset of $\mathcal {G}$ if $\mathcal {D}$ does not have any accumulation points in $\mathcal {G}$. If g is a complex-valued function meromorphic in $\mathcal {G}$, then we use $\mathbb {H}\left( {g}\right)$ to denote the set of all points at which g is holomorphic and we have $\mathcal {Z}\left( {g}\right) =\left\{ {w\in \mathbb {H}\left( {g}\right) }:{g\left( {w}\right) =0}\right\}$. A ${p\times q}$ matrix-valued function $G=\left[ {g_{jk}} \right] _{\begin{array}{c} j=1,\dotsc ,p\\ k=1,\dotsc ,q \end{array}}$ is called meromorphic in $\mathcal {G}$ if $g_{jk}$ is meromorphic in $\mathcal {G}$ for each $j\in \mathbb {Z}_{1,p}$ and each $k\in \mathbb {Z}_{1,q}$. In this case, let $\mathbb {H}\left( {G}\right) :=\bigcap _{j=1}^p\bigcap _{k=1}^q\mathbb {H}\left( {g_{jk}}\right)$.

Let $\Pi _{\mathord {+}}:=\left\{ {z\in \mathbb {C}}:{\Im z\in (0,\infty )}\right\}$ and $\Pi _{\mathord {-}}:=\left\{ {z\in \mathbb {C}}:{\Im z\in (-\infty ,0)}\right\}$.

3 Herglotz–Nevanlinna Functions

In this section, we state some aspects concerning matrix-valued Herglotz–Nevanlinna functions, studied in detail in [9, 17]. The class $\mathcal {R}_{q}\left( {\Pi _{\mathord {+}}}\right)$ of ${q\times q}$ Herglotz–Nevanlinna functions in $\Pi _{\mathord {+}}$ consists of all holomorphic matrix-valued functions $F:\Pi _{\mathord {+}}\rightarrow \mathbb {C}^{{q\times q}}$ satisfying $\Im F\left( {z}\right) \in \mathbb {C}_\succcurlyeq ^{{q\times q}}$ for all $z\in \Pi _{\mathord {+}}$.

Remark 3.1

(cf. [9, Rem. 3.5]) If $F\in \mathcal {R}_{q}\left( {\Pi _{\mathord {+}}}\right)$, then $\mathcal {R}\left( {\left[ {F\left( {z}\right) } \right] ^*}\right) =\mathcal {R}\left( {F\left( {z}\right) }\right)$ and $\mathcal {N}\left( {\left[ {F\left( {z}\right) } \right] ^*}\right) =\mathcal {N}\left( {F\left( {z}\right) }\right)$ for all $z\in \Pi _{\mathord {+}}$.

Denote by $\mathcal {R}_{0,q}\left( {\Pi _{\mathord {+}}}\right)$ the set of all $F\in \mathcal {R}_{q}\left( {\Pi _{\mathord {+}}}\right)$ fulfilling $\sup _{y\in [1,\infty )} y\Vert {F\left( {\textrm{i}y}\right) } \Vert <\infty$.

Theorem 3.2

• (a) For each $F\in \mathcal {R}_{0,q}\left( {\Pi _{\mathord {+}}}\right)$, there exists a unique $\sigma \in {\mathcal {M}_\succcurlyeq ^{q}(\mathbb {R})}$ such that

  $$\begin{aligned} F\left( {z}\right) =\int _\mathbb {R}\frac{1}{x -z}\sigma \left( {\textrm{d}x}\right) \quad \text {for all }z\in \Pi _{\mathord {+}}. \end{aligned}$$

  (3.1)

• (b) If $\sigma \in {\mathcal {M}_\succcurlyeq ^{q}(\mathbb {R})}$, then $F:\Pi _{\mathord {+}}\rightarrow \mathbb {C}^{{q\times q}}$ defined by (3.1) belongs to $\mathcal {R}_{0,q}\left( {\Pi _{\mathord {+}}}\right)$.

If $F\in \mathcal {R}_{0,q}\left( {\Pi _{\mathord {+}}}\right)$, then the unique $\sigma \in {\mathcal {M}_\succcurlyeq ^{q}(\mathbb {R})}$ fulfilling (3.1) is called the $\mathbb {R}$-spectral measure of F. If $\sigma \in {\mathcal {M}_\succcurlyeq ^{q}(\mathbb {R})}$, then $F:\Pi _{\mathord {+}}\rightarrow \mathbb {C}^{{q\times q}}$ defined by (3.1) is said to be the $\mathbb {R}$-Stieltjes transform of $\sigma$.

Remark 3.3

In view of Theorem 3.2, now we can reformulate Problem ${\textsf{MP}[\mathbb {R};(s_j)_{j=0}^{2n},\preccurlyeq ]}$ in the language of $\mathbb {R}$-Stieltjes transforms:

Problem

${\textsf{IP}[\Pi _{\mathord {+}};(s_j)_{j=0}^{2n},\preccurlyeq ]}$ Let $n\in \mathbb {N}_0$ and let $(s_j)_{j=0}^{2n}$ be a sequence of complex ${q\times q}$ matrices. Parametrize the set ${\mathcal {R}_{0,q}\left[ {\Pi _{\mathord {+}};(s_j)_{j=0}^{2n},\preccurlyeq } \right] }$ of all matrix-valued functions $F\in \mathcal {R}_{0,q}\left( {\Pi _{\mathord {+}}}\right)$ the $\mathbb {R}$ spectral measures of which belong to ${\mathcal {M}^{q}_\succcurlyeq [\mathbb {R};(s_j)_{j=0}^{2n},\preccurlyeq ]}$.

It is well known that ${\mathcal {R}_{0,q}\left[ {\Pi _{\mathord {+}};(s_j)_{j=0}^{2n},\preccurlyeq } \right] }\ne \emptyset$ if and only if $(s_j)_{j=0}^{2n}\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,2n}$ (see [1, Thm. 3.2], [21, Satz 9.20], and [4, Thm. 4.16] for (different) proofs in connection with Theorem 3.2).

Lemma 3.4

(cf. [9, Lem. 8.2]) Let $F\in \mathcal {R}_{0,q}\left( {\Pi _{\mathord {+}}}\right)$ with $\mathbb {R}$-spectral measure $\sigma$. For all $z\in \Pi _{\mathord {+}}$, then $\mathcal {R}\left( {F\left( {z}\right) }\right) =\mathcal {R}\left( {\sigma \left( {\mathbb {R}}\right) }\right)$ and $\mathcal {N}\left( {F\left( {z}\right) }\right) =\mathcal {N}\left( {\sigma \left( {\mathbb {R}}\right) }\right)$.

Remark 3.5

Let $F\in \mathcal {R}_{0,q}\left( {\Pi _{\mathord {+}}}\right)$ with $\mathbb {R}$-spectral measure $\sigma$ and let $\widehat{F}:\mathbb {C}\backslash \mathbb {R}\rightarrow \mathbb {C}^{{q\times q}}$ be defined by

$$\begin{aligned} \widehat{F}\left( {z}\right) :={\left\{ \begin{array}{ll} F\left( {z}\right) ,&{}\text { if }z\in \Pi _{\mathord {+}}\\ \left[ {F\left( {\overline{z}}\right) } \right] ^*,&{}\text { if }z\in \Pi _{\mathord {-}} \end{array}\right. }. \end{aligned}$$

(3.2)

Then it is readily checked that $\widehat{F}\left( {z}\right) =\int _\mathbb {R}\left( {x-z}\right) ^{-1}\sigma \left( {\textrm{d}x}\right)$ for all $z\in \mathbb {C}\backslash \mathbb {R}$ (see, e. g. [21, Satz 3.37]).

Lemma 3.6

Let $F\in \mathcal {R}_{0,q}\left( {\Pi _{\mathord {+}}}\right)$ with $\mathbb {R}$-spectral measure $\sigma$ and let $\widehat{F}:\mathbb {C}\backslash \mathbb {R}\rightarrow \mathbb {C}^{{q\times q}}$ be defined by (3.2). For all $z\in \mathbb {C}\backslash \mathbb {R}$, then $\left[ {\widehat{F}\left( {z}\right) } \right] ^*\left[ {\left( {\Im z}\right) ^{-1}\Im \widehat{F}\left( {z}\right) } \right] ^{\mathord {+}}\widehat{F}\left( {z}\right) \preccurlyeq \sigma (\mathbb {R})$ and $\widehat{F}\left( {z}\right) \left[ {\sigma (\mathbb {R})} \right] ^{\mathord {+}}\left[ {\widehat{F}\left( {z}\right) } \right] ^*\preccurlyeq \left( {\Im z}\right) ^{-1}\Im \widehat{F}\left( {z}\right)$.

Proof

Obviously, we have $\sigma \in {\mathcal {M}_\succcurlyeq ^{q}(\mathbb {R})}$. Furthermore, according to Remark 3.5, the function $\widehat{F}$ admits the representation $\widehat{F}\left( {z}\right) =\int _\mathbb {R}\left( {x-z}\right) ^{-1}\sigma \left( {\textrm{d}x}\right)$ for all $z\in \mathbb {C}\backslash \mathbb {R}$. Hence, the assertion is an immediate consequence of [14, Lem. C.7]. $\square$

Lemma 3.7

Let $s_{0}\in \mathbb {C}_\succcurlyeq ^{{q\times q}}$, let $F\in {\mathcal {R}_{0,q}\left[ {\Pi _{\mathord {+}};(s_j)_{j=0}^{0},\preccurlyeq } \right] }$, and let $\widehat{F}:\mathbb {C}\backslash \mathbb {R}\rightarrow \mathbb {C}^{{q\times q}}$ be defined by (3.2). For all $z\in \mathbb {C}\backslash \mathbb {R}$, then $\widehat{F}\left( {z}\right) s_{0}^{\mathord {+}}\left[ {\widehat{F}\left( {z}\right) } \right] ^*\preccurlyeq \left( {\Im z}\right) ^{-1}\Im \widehat{F}\left( {z}\right)$.

Proof

We consider an arbitrary $z\in \mathbb {C}\backslash \mathbb {R}$. Remark 3.3 provides $F\in \mathcal {R}_{0,q}\left( {\Pi _{\mathord {+}}}\right)$ and that the $\mathbb {R}$-spectral measure $\sigma$ of F belongs to ${\mathcal {M}^{q}_\succcurlyeq [\mathbb {R};(s_j)_{j=0}^{0},\preccurlyeq ]}$. Let $M:=\sigma \left( {\mathbb {R}}\right)$. First observe that $\mathcal {R}\left( {\left[ {\widehat{F}\left( {z}\right) } \right] ^*}\right) =\mathcal {R}\left( {M}\right)$ holds true. Indeed, using (3.2), Remark 3.1, and Lemma 3.4, we obtain $\mathcal {R}\left( {\left[ {\widehat{F}\left( {z}\right) } \right] ^*}\right) =\mathcal {R}\left( {\left[ {F\left( {z}\right) } \right] ^*}\right) =\mathcal {R}\left( {F\left( {z}\right) }\right) =\mathcal {R}\left( {M}\right)$ in the case $z\in \Pi _{\mathord {+}}$, whereas (3.2) and Lemma 3.4 yield $\mathcal {R}\left( {\left[ {\widehat{F}\left( {z}\right) } \right] ^*}\right) =\mathcal {R}\left( {F\left( {\overline{z}}\right) }\right) =\mathcal {R}\left( {M}\right)$ in the case $z\in \Pi _{\mathord {-}}$. Setting $P:=\mathbb {P}_{\mathcal {R}\left( {M}\right) }$, hence $P\left[ {\widehat{F}\left( {z}\right) } \right] ^*=\left[ {\widehat{F}\left( {z}\right) } \right] ^*$ follows. Since Remark A.8 shows $P^*=P$, we can then conclude $\widehat{F}\left( {z}\right) P=\widehat{F}\left( {z}\right)$. In view of $\sigma \in {\mathcal {M}^{q}_\succcurlyeq [\mathbb {R};(s_j)_{j=0}^{0},\preccurlyeq ]}$, we have $O_{{q\times q}}\preccurlyeq M\preccurlyeq s_{0}$. Thus, we can apply Lemma A.11 to get $Ps_{0}^{\mathord {+}}P\preccurlyeq M^{\mathord {+}}$. Regarding Remark A.6, then $\widehat{F}\left( {z}\right) Ps_{0}^{\mathord {+}}P\left[ {\widehat{F}\left( {z}\right) } \right] ^*\preccurlyeq \widehat{F}\left( {z}\right) M^{\mathord {+}}\left[ {\widehat{F}\left( {z}\right) } \right] ^*$ follows. From Lemma 3.6 we obtain furthermore $\widehat{F}\left( {z}\right) M^{\mathord {+}}\left[ {\widehat{F}\left( {z}\right) } \right] ^*\preccurlyeq \left( {\Im z}\right) ^{-1}\Im \widehat{F}\left( {z}\right)$. Summarizing, we get $\widehat{F}\left( {z}\right) s_{0}^{\mathord {+}}\left[ {\widehat{F}\left( {z}\right) } \right] ^*=\widehat{F}\left( {z}\right) Ps_{0}^{\mathord {+}}P\left[ {\widehat{F}\left( {z}\right) } \right] ^*\preccurlyeq \widehat{F}\left( {z}\right) M^{\mathord {+}}\left[ {\widehat{F}\left( {z}\right) } \right] ^*\preccurlyeq \left( {\Im z}\right) ^{-1}\Im \widehat{F}\left( {z}\right)$. $\square$

4 Nevanlinna Pairs in $\Pi _{\mathord {+}}$ and Nevanlinna Pairs in $\mathbb {C}\backslash \mathbb {R}$

In this section, we turn our attention to well-known classes of meromorphic matrix-valued functions, which can be used for certain parametrizations of the solution set of matricial power moment problems.

Remark 4.1

The matrix $\tilde{J}_{q}$ given by

$$\begin{aligned} \tilde{J}_{q}:=\begin{bmatrix} O_{{q\times q}}&{}-\textrm{i}I_{q}\\ \textrm{i}I_{q}&{}O_{{q\times q}}\end{bmatrix} \end{aligned}$$

(4.1)

is a ${2q\times 2q}$ signature matrix, i. e., $\tilde{J}_{q}^*=\tilde{J}_{q}$ and $\tilde{J}_{q}^2=I_{2q}$ hold true. Moreover, $\bigl [{\begin{matrix}A \\ B\end{matrix}}\bigr ]^*\left( {-\tilde{J}_{q}}\right) \bigl [{\begin{matrix}A \\ B\end{matrix}}\bigr ]=2\Im \left( {B^*A}\right)$ for all $A,B\in \mathbb {C}^{{q\times q}}$. In particular, the case $B=I_{q}$ is of interest.

Let us recall a well-known notion:

Definition 4.2

Let $\phi$ and $\psi$ be ${q\times q}$ matrix-valued functions meromorphic in $\Pi _{\mathord {+}}$. The pair $\left( {\phi };{\psi }\right)$ is called ${q\times q}$ Nevanlinna pair in $\Pi _{\mathord {+}}$ if there is a discrete subset $\mathcal {D}$ of $\Pi _{\mathord {+}}$ such that the following three conditions are fulfilled:

• (I) $\phi$ and $\psi$ are holomorphic in $\Pi _{\mathord {+}}\backslash \mathcal {D}$.

• (II) ${{\,\textrm{rank}\,}}\left[ \begin{array}{l} \phi \left( {z}\right) \\ \psi \left( {z}\right) \end{array}\right] =q$ for all $z\in \Pi _{\mathord {+}}\backslash \mathcal {D}$.

• (III) $\left[ \begin{array}{l}\phi \left( {z}\right) \\ \psi \left( {z}\right) \end{array}\right] ^*\left( {-\tilde{J}_{q}}\right) \left[ \begin{array}{l}\phi \left( {z}\right) \\ \psi \left( {z}\right) \end{array}\right] \in \mathbb {C}_\succcurlyeq ^{{q\times q}}$ for all $z\in \Pi _{\mathord {+}}\backslash \mathcal {D}$.

We denote the set of all ${q\times q}$ Nevanlinna pairs in $\Pi _{\mathord {+}}$ by $\mathcal{P}\mathcal{R}_{q}\left( {\Pi _{\mathord {+}}}\right)$. Furthermore, let $\mathscr {D}\left( {\phi ,\psi }\right)$ be the set of all discrete subsets $\mathcal {D}$ of $\Pi _{\mathord {+}}$ for which the conditions (I)–(III) hold true.

Remark 4.3

([6, Rem. 4.3]) Remark 4.1 shows that condition (III) in Definition 4.2 is equivalent to:

• (III’) $\Im \left( {\left[ {\psi \left( {z}\right) } \right] ^*\phi \left( {z}\right) }\right) \in \mathbb {C}_\succcurlyeq ^{{q\times q}}$ for all $z\in \Pi _{\mathord {+}}\backslash \mathcal {D}$.

Remark 4.4

(cf. [6, Rem. 4.4]) Let $\left( {\phi };{\psi }\right) \in \mathcal{P}\mathcal{R}_{q}\left( {\Pi _{\mathord {+}}}\right)$. For each ${q\times q}$ matrix-valued function $g$ meromorphic in $\Pi _{\mathord {+}}$ such that $\det g$ does not vanish identically, it is readily checked that the pair $\left( {\phi g};{\psi g}\right)$ belongs to $\mathcal{P}\mathcal{R}_{q}\left( {\Pi _{\mathord {+}}}\right)$ as well. Two pairs $\left( {\phi _1};{\psi _1}\right)$ and $\left( {\phi _2};{\psi _2}\right)$ belonging to $\mathcal{P}\mathcal{R}_{q}\left( {\Pi _{\mathord {+}}}\right)$ are said to be equivalent if there exists a ${q\times q}$ matrix-valued function $g$ meromorphic in $\Pi _{\mathord {+}}$ such that $\det g$ does not vanish identically, satisfying $\phi _2=\phi _1g$ and $\psi _2=\psi _1g$. Indeed, it is readily checked that this relation is an equivalence relation on $\mathcal{P}\mathcal{R}_{q}\left( {\Pi _{\mathord {+}}}\right)$. For each $\left( {\phi };{\psi }\right) \in \mathcal{P}\mathcal{R}_{q}\left( {\Pi _{\mathord {+}}}\right)$, we denote by $\langle {\left( {\phi };{\psi }\right) } \rangle$ the equivalence class generated by $\left( {\phi };{\psi }\right)$.

Notation 4.5

If $M\in \mathbb {C}^{{q\times p}}$, then let $\mathcal {P}\left[ {M} \right]$ be the set of all $\left( {\phi };{\psi }\right) \in \mathcal{P}\mathcal{R}_{q}\left( {\Pi _{\mathord {+}}}\right)$ satisfying $\mathbb {P}_{\mathcal {R}\left( {M}\right) }\phi =\phi$.

Lemma 4.6

([6, Lem. 4.7]) Let $M\in \mathbb {C}_\textrm{H}^{{q\times q}}$, let $\left( {\phi };{\psi }\right) \in \mathcal {P}\left[ {M} \right]$, and let $P:=\mathbb {P}_{\mathcal {R}\left( {M}\right) }$ and $Q:=\mathbb {P}_{\mathcal {N}\left( {M}\right) }$. Then there exists a pair $\left( {S};{T}\right) \in \langle {\left( {\phi };{\psi }\right) } \rangle$ such that

$$\begin{aligned} PS&=S,&SP&=S,&PT&=T-Q,{} & {} \text {and}&TP&=T-Q. \end{aligned}$$

(4.2)

In Definition 4.2 we recalled the class $\mathcal{P}\mathcal{R}_{q}\left( {\Pi _{\mathord {+}}}\right)$ of all ${q\times q}$ Nevanlinna pairs in $\Pi _{\mathord {+}}$. Now we consider a further class of pairs of meromorphic matrix-valued functions, which is related to the class $\mathcal{P}\mathcal{R}_{q}\left( {\Pi _{\mathord {+}}}\right)$. More precisely, these pairs will come in handy when synchronizing the two matricial Hamburger moment problems which a given matricial Stieltjes moment problem can be reduced to.

Definition 4.7

Let $\eta$ and $\theta$ be ${q\times q}$ matrix-valued functions meromorphic in $\mathbb {C}\backslash \mathbb {R}$. The pair $\left( {\eta };{\theta }\right)$ is called ${q\times q}$ Nevanlinna pair in $\mathbb {C}\backslash \mathbb {R}$ if there exists a discrete subset $\mathcal {E}$ of $\mathbb {C}\backslash \mathbb {R}$ such that the following four conditions are fulfilled:

• (I) $\eta$ and $\theta$ are holomorphic in $\mathbb {C}\backslash \left( {\mathbb {R}\cup \mathcal {E}}\right)$.

• (II) ${{\,\textrm{rank}\,}}\left[ \begin{array}{l} \eta \left( {z}\right) \\ \theta \left( {z}\right) \end{array}\right] =q$ for all $z\in \mathbb {C}\backslash \left( {\mathbb {R}\cup \mathcal {E}}\right)$.

• (III) $\frac{1}{\Im z}\left[ \begin{array}{l}\eta \left( {z}\right) \\ \theta \left( {z}\right) \end{array}\right] ^*\left( {-\tilde{J}_{q}}\right) \left[ \begin{array}{l}\eta \left( {z}\right) \\ \theta \left( {z}\right) \end{array}\right] \succcurlyeq O_{{q\times q}}$ for all $z\in \mathbb {C}\backslash \left( {\mathbb {R}\cup \mathcal {E}}\right)$.

• (IV) $\left[ \begin{array}{l}\eta \left( {\overline{z}}\right) \\ \theta \left( {\overline{z}}\right) \end{array}\right] ^*\left( {-\tilde{J}_{q}}\right) \left[ \begin{array}{l}\eta \left( {z}\right) \\ \theta \left( {z}\right) \end{array}\right] =O_{{q\times q}}$ for all $z\in \mathbb {C}\backslash \left( {\mathbb {R}\cup \mathcal {E}}\right)$.

We denote the set of all ${q\times q}$ Nevanlinna pairs in $\mathbb {C}\backslash \mathbb {R}$ by $\mathcal{P}\mathcal{R}_{q}\left( {\mathbb {C}\backslash \mathbb {R}}\right)$. Furthermore, let $\mathscr {E}\left( {\eta ,\theta }\right)$ be the set of all discrete subsets $\mathcal {E}$ of $\mathbb {C}\backslash \mathbb {R}$ for which the conditions (I)–(IV) hold true.

Remark 4.8

Let $\left( {\eta };{\theta }\right) \in \mathcal{P}\mathcal{R}_{q}\left( {\mathbb {C}\backslash \mathbb {R}}\right)$ and let $\mathcal {E}\in \mathscr {E}\left( {\eta ,\theta }\right)$. Denote by $\phi$ and $\psi$ the restrictions of $\eta$ and $\theta$ onto $\Pi _{\mathord {+}}$, respectively, and let $\mathcal {D}:=\mathcal {E}\cap \Pi _{\mathord {+}}$. Regarding Definitions 4.7 and 4.2, then $\left( {\phi };{\psi }\right) \in \mathcal{P}\mathcal{R}_{q}\left( {\Pi _{\mathord {+}}}\right)$ and $\mathcal {D}\in \mathscr {D}\left( {\phi ,\psi }\right)$.

Remark 4.9

• (a) Remark 4.1 shows that condition (III) in Definition 4.7 can be replaced equivalently by the following condition (III’):

  • (III’) $\left( {\Im z}\right) ^{-1}\Im \left( {\left[ {\theta \left( {z}\right) } \right] ^*\eta \left( {z}\right) }\right) \in \mathbb {C}_\succcurlyeq ^{{q\times q}}$ for all $z\in \mathbb {C}\backslash \left( {\mathbb {R}\cup \mathcal {E}}\right)$.

• (b) In view of (4.1), it is readily checked that condition (IV) in Definition 4.7 can be replaced equivalently by the following condition (IV’):

  • (IV’) $\left[ {\eta \left( {\overline{z}}\right) } \right] ^*\theta \left( {z}\right) =\left[ {\theta \left( {\overline{z}}\right) } \right] ^*\eta \left( {z}\right)$ for all $z\in \mathbb {C}\backslash \left( {\mathbb {R}\cup \mathcal {E}}\right)$.

Notation 4.10

If $M\in \mathbb {C}^{{q\times p}}$, then let $\hat{\mathcal {P}}\left[ {M} \right]$ be the set of all $\left( {\eta };{\theta }\right) \in \mathcal{P}\mathcal{R}_{q}\left( {\mathbb {C}\backslash \mathbb {R}}\right)$ satisfying $\mathbb {P}_{\mathcal {R}\left( {M}\right) }\eta =\eta$.

Lemma 4.11

Let $\left( {\phi };{\psi }\right) \in \mathcal{P}\mathcal{R}_{q}\left( {\Pi _{\mathord {+}}}\right)$ and let $\mathcal {D}\in \mathscr {D}\left( {\phi ,\psi }\right)$. Then:

• (a) The set $\mathcal {D}$ is a discrete subset of $\Pi _{\mathord {+}}$ and both matrix-valued functions $\phi$ and $\psi$ are meromorphic in $\Pi _{\mathord {+}}$ and holomorphic in $\Pi _{\mathord {+}}\backslash \mathcal {D}$.

• (b) Let $\tilde{\epsilon }:\Pi _{\mathord {+}}\rightarrow \mathbb {C}$ be defined by $\tilde{\epsilon }\left( {z}\right) :=z$. Then

  $$\begin{aligned} R :=\phi +\tilde{\epsilon }\psi \end{aligned}$$

  (4.3)

  is meromorphic in $\Pi _{\mathord {+}}$ and holomorphic in $\Pi _{\mathord {+}}\backslash \mathcal {D}$, fulfilling $\det R\left( {z}\right) \ne 0$ for all $z\in \Pi _{\mathord {+}}\backslash \mathcal {D}$.

• (c) The set $\mathcal {D}^{\mathord {\vee }}$ is a discrete subset of $\Pi _{\mathord {-}}$ and both matrix-valued functions $\left( {\phi R^{-1}}\right) ^{\mathord {\vee }}$ and $\left( {\psi R^{-1}}\right) ^{\mathord {\vee }}$ are meromorphic in $\Pi _{\mathord {-}}$ and holomorphic in $\Pi _{\mathord {-}}\backslash \mathcal {D}^{\mathord {\vee }}$.

• (d) The set $\mathcal {D}_{\mathord {\diamond }}:=\mathcal {D}\cup \mathcal {D}^{\mathord {\vee }}$ is a discrete subset of $\mathbb {C}\backslash \mathbb {R}$ and both matrix-valued functions

  $$\begin{aligned} \phi _{\mathord {\diamond }}&:={\left\{ \begin{array}{ll} \phi &{}\text { in }\Pi _{\mathord {+}}\\ \left( {\phi R^{-1}}\right) ^{\mathord {\vee }}&{}\text { in }\Pi _{\mathord {-}}\end{array}\right. }{} & {} \text {and}&\psi _{\mathord {\diamond }}&:={\left\{ \begin{array}{ll} \psi &{}\text { in }\Pi _{\mathord {+}}\\ \left( {\psi R^{-1}}\right) ^{\mathord {\vee }}&{}\text { in }\Pi _{\mathord {-}}\end{array}\right. }\nonumber \\ \end{aligned}$$

  (4.4)

  are meromorphic in $\mathbb {C}\backslash \mathbb {R}$ and holomorphic in $\mathbb {C}\backslash \left( {\mathbb {R}\cup \mathcal {D}_{\mathord {\diamond }}}\right)$, fulfilling

  $$\begin{aligned} \phi _{\mathord {\diamond }}\left( {z}\right)&=\phi \left( {z}\right) ,&\psi _{\mathord {\diamond }}\left( {z}\right)&=\psi \left( {z}\right)&\text {for all }z&\in \Pi _{\mathord {+}}\backslash \mathcal {D} \end{aligned}$$

  (4.5)

  and

  $$\begin{aligned} \phi _{\mathord {\diamond }}\left( {w}\right) =\left[ {R\left( {\overline{w}}\right) } \right] ^{-*}\left[ {\phi \left( {\overline{w}}\right) } \right] ^*,\; \psi _{\mathord {\diamond }}\left( {w}\right) =\left[ {R\left( {\overline{w}}\right) } \right] ^{-*}\left[ {\psi \left( {\overline{w}}\right) } \right] ^*\text { for all }w\in \Pi _{\mathord {-}}\backslash \mathcal {D}^{\mathord {\vee }}.\nonumber \\ \end{aligned}$$

  (4.6)

• (e) The pair $\left( {\phi _{\mathord {\diamond }}};{\psi _{\mathord {\diamond }}}\right)$ belongs to $\mathcal{P}\mathcal{R}_{q}\left( {\mathbb {C}\backslash \mathbb {R}}\right)$ and $\mathcal {D}_{\mathord {\diamond }}\in \mathscr {E}\left( {\phi _{\mathord {\diamond }},\psi _{\mathord {\diamond }}}\right)$.

Proof

By virtue of Definition 4.2, obviously part (a) is fulfilled.

Taking into account (4.3), part (a), and that the function $\tilde{\epsilon }$ is holomorphic in $\Pi _{\mathord {+}}$, we infer that R is meromorphic in $\Pi _{\mathord {+}}$ and holomorphic in $\Pi _{\mathord {+}}\backslash \mathcal {D}$. Set $s_{0}:=I_{q}$. Then $\mathbb {P}_{\mathcal {R}\left( {s_{0}}\right) }\phi =\phi$. According to Notation 4.5, then $\left( {\phi };{\psi }\right) \in \mathcal {P}\left[ {s_{0}} \right]$ follows. Let $V:\Pi _{\mathord {+}}\rightarrow \mathbb {C}^{{2q\times 2q}}$ be defined by $V\left( {z}\right) :=\left[ \begin{array}{cc}O_{{q\times q}}&{} -s_{0}\\ s_{0}^{\mathord {+}}&{} zI_{q}\end{array}\right]$. Regarding (4.3), we get then

$$\begin{aligned} V \begin{bmatrix}\phi \\ \psi \end{bmatrix} =\begin{bmatrix}O_{{q\times q}}&{}-I_{q}\\ I_{q}&{}\tilde{\epsilon }I_{q}\end{bmatrix} \begin{bmatrix}\phi \\ \psi \end{bmatrix} =\begin{bmatrix}-\psi \\ \phi +\tilde{\epsilon }\psi \end{bmatrix} =\begin{bmatrix}-\psi \\ R\end{bmatrix}. \end{aligned}$$

The application of [5, Prop.  8.8] thus yields $\det R\left( {z}\right) \ne 0$ for all $z\in \Pi _{\mathord {+}}\backslash \mathcal {D}$ and that $F:=-\psi R^{-1}$ belongs to ${\mathcal {R}_{0,q}\left[ {\Pi _{\mathord {+}};(s_j)_{j=0}^{0},\preccurlyeq } \right] }$. In particular, part (b) is proved.

In view of (2.1) it is readily checked that $\mathcal {D}^{\mathord {\vee }}$ is a discrete subset of $\Pi _{\mathord {-}}$. From part (b) we can infer that $R^{-1}$ is meromorphic in $\Pi _{\mathord {+}}$ and holomorphic in $\Pi _{\mathord {+}}\backslash \mathcal {D}$. Regarding part (a), consequently $\phi R^{-1}$ and $\psi R^{-1}$ are meromorphic in $\Pi _{\mathord {+}}$ and holomorphic in $\Pi _{\mathord {+}}\backslash \mathcal {D}$. Taking into account (2.1) and (2.2), it is then readily checked that $\left( {\phi R^{-1}}\right) ^{\mathord {\vee }}$ and $\left( {\psi R^{-1}}\right) ^{\mathord {\vee }}$ are meromorphic in $\Pi _{\mathord {-}}$ and holomorphic in $\Pi _{\mathord {-}}\backslash \mathcal {D}^{\mathord {\vee }}$. Hence, part (c) is checked.

By virtue of parts (a) and (c), we can infer that $\mathcal {D}_{\mathord {\diamond }}$ is a discrete subset of $\mathbb {C}\backslash \mathbb {R}$ and that $\phi _{\mathord {\diamond }}$ and $\psi _{\mathord {\diamond }}$ are meromorphic in $\mathbb {C}\backslash \mathbb {R}$ and holomorphic in $\mathbb {C}\backslash \left( {\mathbb {R}\cup \mathcal {D}_{\mathord {\diamond }}}\right)$. In view of part (a), the identities (4.5) are an immediate consequence of (4.4). Now we consider an arbitrary $w\in \Pi _{\mathord {-}}\backslash \mathcal {D}^{\mathord {\vee }}$. Regarding (2.1), then $\overline{w}\in \Pi _{\mathord {+}}\backslash \mathcal {D}$. Taking into account parts (a)–(c) as well as (2.2) and (4.4), we can conclude $\det R\left( {\overline{w}}\right) \ne 0$ as well as $\left[ {R\left( {\overline{w}}\right) } \right] ^{-*}\left[ {\phi \left( {\overline{w}}\right) } \right] ^*=\left( {\phi \left( {\overline{w}}\right) \left[ {R\left( {\overline{w}}\right) } \right] ^{-1}}\right) ^*=\left[ {\left( {\phi R}\right) ^{-1}\left( {\overline{w}}\right) } \right] ^*=\left( {\phi R}\right) ^{\mathord {\vee }}\left( {w}\right) =\phi _{\mathord {\diamond }}\left( {w}\right)$ and similarly $\left[ {R\left( {\overline{w}}\right) } \right] ^{-*}\left[ {\psi \left( {\overline{w}}\right) } \right] ^*=\psi _{\mathord {\diamond }}\left( {w}\right)$. Consequently, part (d) is proved. Now we are going to check that

$$\begin{aligned} {{\,\textrm{rank}\,}}\begin{bmatrix}\phi _{\mathord {\diamond }}\left( {z}\right) \\ \psi _{\mathord {\diamond }}\left( {z}\right) \end{bmatrix}&=q,&\frac{1}{\Im z}\Im \left( {\left[ {\psi _{\mathord {\diamond }}\left( {z}\right) } \right] ^*\phi _{\mathord {\diamond }}\left( {z}\right) }\right)&\in \mathbb {C}_\succcurlyeq ^{{q\times q}},&\end{aligned}$$

(4.7)

and

$$\begin{aligned} \left[ {\phi _{\mathord {\diamond }}\left( {\overline{z}}\right) } \right] ^*\psi _{\mathord {\diamond }}\left( {z}\right) =\left[ {\psi _{\mathord {\diamond }}\left( {\overline{z}}\right) } \right] ^*\phi _{\mathord {\diamond }}\left( {z}\right) \end{aligned}$$

(4.8)

hold true for all $z\in \mathbb {C}\backslash \left( {\mathbb {R}\cup \mathcal {D}_{\mathord {\diamond }}}\right)$. First we consider an arbitrary $z\in \Pi _{\mathord {+}}\backslash \mathcal {D}$. Using (4.5), Definition 4.2, and Remark 4.3, we can infer then (4.7). In view of parts (b) and (a), we have furthermore $\det R\left( {z}\right) \ne 0$ and $R\left( {z}\right) =\phi \left( {z}\right) +z\psi \left( {z}\right)$. Hence, we get $\phi \left( {z}\right) \left[ {R\left( {z}\right) } \right] ^{-1}\psi \left( {z}\right) =\left[ {R\left( {z}\right) -z\psi \left( {z}\right) } \right] \left[ {R\left( {z}\right) } \right] ^{-1}\psi \left( {z}\right) =\psi \left( {z}\right) -z\psi \left( {z}\right) \left[ {R\left( {z}\right) } \right] ^{-1}\psi \left( {z}\right)$ and $\psi \left( {z}\right) \left[ {R\left( {z}\right) } \right] ^{-1}\phi \left( {z}\right) =\psi \left( {z}\right) \left[ {R\left( {z}\right) } \right] ^{-1}\left[ {R\left( {z}\right) -z\psi \left( {z}\right) } \right] =\psi \left( {z}\right) -z\psi \left( {z}\right) \left[ {R\left( {z}\right) } \right] ^{-1}\psi \left( {z}\right)$, implying

$$\begin{aligned} \phi \left( {z}\right) \left[ {R\left( {z}\right) } \right] ^{-1}\psi \left( {z}\right) =\psi \left( {z}\right) \left[ {R\left( {z}\right) } \right] ^{-1}\phi \left( {z}\right) . \end{aligned}$$

(4.9)

Regarding (2.1), we see $\overline{z}\in \Pi _{\mathord {-}}\backslash \mathcal {D}^{\mathord {\vee }}$. Taking additionally into account (4.6) and (4.5), we can conclude $\left[ {\phi _{\mathord {\diamond }}\left( {\overline{z}}\right) } \right] ^*\psi _{\mathord {\diamond }}\left( {z}\right) =\left( {\left[ {R\left( {z}\right) } \right] ^{-*}\left[ {\phi \left( {z}\right) } \right] ^*}\right) ^*\psi \left( {z}\right) =\phi \left( {z}\right) \left[ {R\left( {z}\right) } \right] ^{-1}\psi \left( {z}\right)$ and $\left[ {\psi _{\mathord {\diamond }}\left( {\overline{z}}\right) } \right] ^*\phi _{\mathord {\diamond }}\left( {z}\right) =\left( {\left[ {R\left( {z}\right) } \right] ^{-*}\left[ {\psi \left( {z}\right) } \right] ^*}\right) ^*\phi \left( {z}\right) =\psi \left( {z}\right) \left[ {R\left( {z}\right) } \right] ^{-1}\phi \left( {z}\right)$. In view of (4.9), then $\left[ {\phi _{\mathord {\diamond }}\left( {\overline{z}}\right) } \right] ^*\psi _{\mathord {\diamond }}\left( {z}\right) =\left[ {\psi _{\mathord {\diamond }}\left( {\overline{z}}\right) } \right] ^*\phi _{\mathord {\diamond }}\left( {z}\right)$ follows. Therefore, (4.7) and (4.8) are checked for all $z\in \Pi _{\mathord {+}}\backslash \mathcal {D}$. Now we consider an arbitrary $w\in \Pi _{\mathord {-}}\backslash \mathcal {D}^{\mathord {\vee }}$. Regarding (2.1), then $\overline{w}\in \Pi _{\mathord {+}}\backslash \mathcal {D}$. Hence, in view of parts (b) and (a), we have

$$\begin{aligned} \det R\left( {\overline{w}}\right)&\ne 0,&R\left( {\overline{w}}\right)&=\phi \left( {\overline{w}}\right) +\overline{w}\psi \left( {\overline{w}}\right) ,{} & {} \text {and}&F\left( {\overline{w}}\right)&=-\psi \left( {\overline{w}}\right) \left[ {R\left( {\overline{w}}\right) } \right] ^{-1}. \end{aligned}$$

(4.10)

Taking additionally into account (4.6), we can conclude

$$\begin{aligned}\begin{aligned} q&={{\,\textrm{rank}\,}}\left( {\left[ {R\left( {\overline{w}}\right) } \right] ^{-*}\left[ {R\left( {\overline{w}}\right) } \right] ^*}\right) ={{\,\textrm{rank}\,}}\left( {\left[ {R\left( {\overline{w}}\right) } \right] ^{-*}\left( {\left[ {\phi \left( {\overline{w}}\right) } \right] ^*+w\left[ {\psi \left( {\overline{w}}\right) } \right] ^*}\right) }\right) \\&={{\,\textrm{rank}\,}}\left( {\left[ {I_{q},wI_{q}} \right] \begin{bmatrix}\left[ {R\left( {\overline{w}}\right) } \right] ^{-*}\left[ {\phi \left( {\overline{w}}\right) } \right] ^*\\ \left[ {R\left( {\overline{w}}\right) } \right] ^{-*}\left[ {\psi \left( {\overline{w}}\right) } \right] ^*\end{bmatrix}}\right) \le {{\,\textrm{rank}\,}}\begin{bmatrix}\left[ {R\left( {\overline{w}}\right) } \right] ^{-*}\left[ {\phi \left( {\overline{w}}\right) } \right] ^*\\ \left[ {R\left( {\overline{w}}\right) } \right] ^{-*}\left[ {\psi \left( {\overline{w}}\right) } \right] ^*\end{bmatrix}\\&={{\,\textrm{rank}\,}}\begin{bmatrix}\phi _{\mathord {\diamond }}\left( {w}\right) \\ \psi _{\mathord {\diamond }}\left( {w}\right) \end{bmatrix} \le q. \end{aligned}\end{aligned}$$

Consequently, ${{\,\textrm{rank}\,}}\bigl [{\begin{matrix}\phi _{\mathord {\diamond }}\left( {w}\right) \\ \psi _{\mathord {\diamond }}\left( {w}\right) \end{matrix}}\bigr ]=q$ follows. In view of (4.6) and (4.10), we see that

$$\begin{aligned}\begin{aligned} \phi _{\mathord {\diamond }}\left( {w}\right)&=\left[ {R\left( {\overline{w}}\right) } \right] ^{-*}\left[ {\phi \left( {\overline{w}}\right) } \right] ^*=\left[ {R\left( {\overline{w}}\right) } \right] ^{-*}\left[ {R\left( {\overline{w}}\right) -\overline{w}\psi \left( {\overline{w}}\right) } \right] ^*\\&=\left( {\left[ {R\left( {\overline{w}}\right) -\overline{w}\psi \left( {\overline{w}}\right) } \right] \left[ {R\left( {\overline{w}}\right) } \right] ^{-1}}\right) ^*=\left( {I_{q}-\overline{w}\psi \left( {\overline{w}}\right) \left[ {R\left( {\overline{w}}\right) } \right] ^{-1}}\right) ^*\\&=I_{q}-w\left( {\psi \left( {\overline{w}}\right) \left[ {R\left( {\overline{w}}\right) } \right] ^{-1}}\right) ^*=I_{q}+w\left[ {F\left( {\overline{w}}\right) } \right] ^*\end{aligned}\end{aligned}$$

and $\left[ {\psi _{\mathord {\diamond }}\left( {w}\right) } \right] ^*=\left( {\left[ {R\left( {\overline{w}}\right) } \right] ^{-*}\left[ {\psi \left( {\overline{w}}\right) } \right] ^*}\right) ^*=\psi \left( {\overline{w}}\right) \left[ {R\left( {\overline{w}}\right) } \right] ^{-1}=-F\left( {\overline{w}}\right)$. Hence, we can conclude

$$\begin{aligned} \left[ {\psi _{\mathord {\diamond }}\left( {w}\right) } \right] ^*\phi _{\mathord {\diamond }}\left( {w}\right) =-F\left( {\overline{w}}\right) \left( {I_{q}+w\left[ {F\left( {\overline{w}}\right) } \right] ^*}\right) =-F\left( {\overline{w}}\right) -wF\left( {\overline{w}}\right) \left[ {F\left( {\overline{w}}\right) } \right] ^*. \end{aligned}$$

Using Remarks A.1 and A.3, thus we obtain

$$\begin{aligned}\begin{aligned} \Im \left( {\left[ {\psi _{\mathord {\diamond }}\left( {w}\right) } \right] ^*\phi _{\mathord {\diamond }}\left( {w}\right) }\right)&=-\Im F\left( {\overline{w}}\right) -\Im \left( {wF\left( {\overline{w}}\right) \left[ {F\left( {\overline{w}}\right) } \right] ^*}\right) \\&=-\Im F\left( {\overline{w}}\right) -\Im \left( {w}\right) F\left( {\overline{w}}\right) \left[ {F\left( {\overline{w}}\right) } \right] ^*. \end{aligned}\end{aligned}$$

Regarding $s_{0}=I_{q}$, then

$$\begin{aligned} \begin{aligned} \left( {\Im w}\right) ^{-1}\Im \left( {\left[ {\psi _{\mathord {\diamond }}\left( {w}\right) } \right] ^*\phi _{\mathord {\diamond }}\left( {w}\right) }\right)&=-\left( {\Im w}\right) ^{-1}\Im F\left( {\overline{w}}\right) -F\left( {\overline{w}}\right) \left[ {F\left( {\overline{w}}\right) } \right] ^*\\&=\left( {\Im \overline{w}}\right) ^{-1}\Im F\left( {\overline{w}}\right) -F\left( {\overline{w}}\right) s_{0}^{\mathord {+}}\left[ {F\left( {\overline{w}}\right) } \right] ^*\end{aligned}\end{aligned}$$

(4.11)

follows. In view of $\overline{w}\in \Pi _{\mathord {+}}$, we see that $\widehat{F}:\mathbb {C}\backslash \mathbb {R}\rightarrow \mathbb {C}^{{q\times q}}$ defined by (3.2) fulfills $\widehat{F}\left( {\overline{w}}\right) =F\left( {\overline{w}}\right)$. Taking additionally into account $s_{0}\in \mathbb {C}_\succcurlyeq ^{{q\times q}}$ and $F\in {\mathcal {R}_{0,q}\left[ {\Pi _{\mathord {+}};(s_j)_{j=0}^{0},\preccurlyeq } \right] }$, we can then infer from Lemma 3.7 that the matrix on the right-hand side in (4.11) is non-negative Hermitian. Hence, (4.11) implies $\left( {\Im w}\right) ^{-1}\Im \left( {\left[ {\psi _{\mathord {\diamond }}\left( {w}\right) } \right] ^*\phi _{\mathord {\diamond }}\left( {w}\right) }\right) \in \mathbb {C}_\succcurlyeq ^{{q\times q}}$. Regarding $\overline{w}\in \Pi _{\mathord {+}}\backslash \mathcal {D}$ and (4.9), we can infer $\phi \left( {\overline{w}}\right) \left[ {R\left( {\overline{w}}\right) } \right] ^{-1}\psi \left( {\overline{w}}\right) =\psi \left( {\overline{w}}\right) \left[ {R\left( {\overline{w}}\right) } \right] ^{-1}\phi \left( {\overline{w}}\right)$. According to (4.5) and (4.6), we have $\left[ {\phi _{\mathord {\diamond }}\left( {\overline{w}}\right) } \right] ^*\psi _{\mathord {\diamond }}\left( {w}\right) =\left[ {\phi \left( {\overline{w}}\right) } \right] ^*\left[ {R\left( {\overline{w}}\right) } \right] ^{-*}\left[ {\psi \left( {\overline{w}}\right) } \right] ^*=\left( {\psi \left( {\overline{w}}\right) \left[ {R\left( {\overline{w}}\right) } \right] ^{-1}\phi \left( {\overline{w}}\right) }\right) ^*$ and $\left[ {\psi _{\mathord {\diamond }}\left( {\overline{w}}\right) } \right] ^*\phi _{\mathord {\diamond }}\left( {w}\right) =\left[ {\psi \left( {\overline{w}}\right) } \right] ^*\left[ {R\left( {\overline{w}}\right) } \right] ^{-*}\left[ {\phi \left( {\overline{w}}\right) } \right] ^*=\left( {\phi \left( {\overline{w}}\right) \left[ {R\left( {\overline{w}}\right) } \right] ^{-1}\psi \left( {\overline{w}}\right) }\right) ^*$. Consequently, $\left[ {\phi _{\mathord {\diamond }}\left( {\overline{w}}\right) } \right] ^*\psi _{\mathord {\diamond }}\left( {w}\right) =\left[ {\psi _{\mathord {\diamond }}\left( {\overline{w}}\right) } \right] ^*\phi _{\mathord {\diamond }}\left( {w}\right)$ follows. Therefore, we have shown that (4.7) and (4.8) are fulfilled for all $z\in \Pi _{\mathord {-}}\backslash \mathcal {D}^{\mathord {\vee }}$ as well. Regarding $\left( {\Pi _{\mathord {+}}\backslash \mathcal {D}}\right) \cup \left( {\Pi _{\mathord {-}}\backslash \mathcal {D}^{\mathord {\vee }}}\right) =\mathbb {C}\backslash \left( {\mathbb {R}\cup \mathcal {D}_{\mathord {\diamond }}}\right)$, we hence have (4.7) and (4.8) for all $z\in \mathbb {C}\backslash \left( {\mathbb {R}\cup \mathcal {D}_{\mathord {\diamond }}}\right)$. Taking additionally into account part (d) and regarding Definition 4.7 and Remark 4.9, we then can conclude $\left( {\phi _{\mathord {\diamond }}};{\psi _{\mathord {\diamond }}}\right) \in \mathcal{P}\mathcal{R}_{q}\left( {\mathbb {C}\backslash \mathbb {R}}\right)$ and $\mathcal {D}_{\mathord {\diamond }}\in \mathscr {E}\left( {\phi _{\mathord {\diamond }},\psi _{\mathord {\diamond }}}\right)$. Thus, part (e) is proved. $\square$

Lemma 4.12

Let $M \in \mathbb {C}_\textrm{H}^{{q\times q}}$ and let $\left( {\phi };{\psi }\right) \in \mathcal {P}\left[ {M} \right]$. Then $\left( {\phi _{\mathord {\diamond }}};{\psi _{\mathord {\diamond }}}\right)$ given by (4.4) belongs to $\hat{\mathcal {P}}\left[ {M} \right]$.

Proof

In view of Notation 4.5, we have $\left( {\phi };{\psi }\right) \in \mathcal{P}\mathcal{R}_{q}\left( {\Pi _{\mathord {+}}}\right)$. According to Definition 4.2, then $\phi$ and $\psi$ are meromorphic in $\Pi _{\mathord {+}}$ and there exists a set $\mathcal {D}\in \mathscr {D}\left( {\phi ,\psi }\right)$. Let $\tilde{\epsilon }:\Pi _{\mathord {+}}\rightarrow \mathbb {C}$ be defined by $\tilde{\epsilon }\left( {z}\right) :=z$, let $R:=\phi +\tilde{\epsilon }\psi$, and let $\left( {\phi _{\mathord {\diamond }}};{\psi _{\mathord {\diamond }}}\right)$ be given by (4.4). Lemma 4.11 then shows that R is meromorphic in $\Pi _{\mathord {+}}$, that $\det R$ does not vanish identically, and that $\left( {\phi _{\mathord {\diamond }}};{\psi _{\mathord {\diamond }}}\right)$ belongs to $\mathcal{P}\mathcal{R}_{q}\left( {\mathbb {C}\backslash \mathbb {R}}\right)$. Now we are going to prove $P\phi _{\mathord {\diamond }}=\phi _{\mathord {\diamond }}$, where $P:=\mathbb {P}_{\mathcal {R}\left( {M}\right) }$. Setting $Q:=\mathbb {P}_{\mathcal {N}\left( {M}\right) }$, the application of Lemma 4.6 yields the existence of a pair $\left( {S};{T}\right) \in \langle {\left( {\phi };{\psi }\right) } \rangle$ satisfying (4.2). Using Remark 4.4, we can infer then that there exists a ${q\times q}$ matrix-valued function $g$ meromorphic in $\Pi _{\mathord {+}}$ such that $\det g$ does not vanish identically, satisfying $S=\phi g$ and $T=\psi g$. Taking additionally into account that (4.2) implies $S=SP$ and $T=TP+Q$, we thus obtain

$$\begin{aligned}{} & {} Rg=\left( {\phi +\tilde{\epsilon }\psi }\right) g=\phi g+\tilde{\epsilon }\psi g=S+\tilde{\epsilon }T\\{} & {} \quad =SP+\tilde{\epsilon }\left( {TP+Q}\right) =\left( {S+\tilde{\epsilon }T}\right) P+\tilde{\epsilon }Q. \end{aligned}$$

In view of $M^*=M$, we see from Remark A.7 that $\mathcal {N}\left( {M}\right) =\mathcal {R}\left( {M}\right) ^\bot$. Consequently, Remark A.9 yields $P+Q=I_{q}$, whereas Remark A.8 provides $\mathcal {R}\left( {Q}\right) =\mathcal {N}\left( {P}\right)$. Hence, $PQ=O_{{q\times q}}$ follows. Remark A.8 furthermore shows $P^*=P$ as well as $Q^2=Q$. Thus, we can conclude

$$\begin{aligned} RgQ =\left[ {\left( {S+\tilde{\epsilon }T}\right) P+\tilde{\epsilon }Q} \right] Q =\left( {S+\tilde{\epsilon }T}\right) PQ+\tilde{\epsilon }Q^2 =\tilde{\epsilon }Q =\tilde{\epsilon }\left( {I_{q}-P}\right) . \end{aligned}$$

Obviously, $\tilde{\epsilon }$ is meromorphic in $\Pi _{\mathord {+}}$ and does not vanish identically. Since $\det R$ and $\det g$ does not vanish identically as well, then $\tilde{\epsilon }^{-1}Q=g^{-1}R^{-1}\left( {I_{q}-P}\right)$ and, hence, $\tilde{\epsilon }^{-1}Q+g^{-1}R^{-1}P=g^{-1}R^{-1}$ follow. Using $PQ=O_{{q\times q}}$, we thus get $Pg^{-1}R^{-1}P=Pg^{-1}R^{-1}$. Since $S=SP$ and $S=\phi g$ imply $SP=\phi g$, we can infer then $\phi R^{-1}P=\phi R^{-1}$. Regarding $P^*=P$ and (2.2), we consequently conclude $\left( {\phi R^{-1}}\right) ^{\mathord {\vee }}=\left( {\phi R^{-1}P}\right) ^{\mathord {\vee }}=P^*\left( {\phi R^{-1}}\right) ^{\mathord {\vee }}=P\left( {\phi R^{-1}}\right) ^{\mathord {\vee }}$. According to Notation 4.5, we have furthermore $P\phi =\phi$. In view of (4.4), therefore $P\phi _{\mathord {\diamond }}=\phi _{\mathord {\diamond }}$ follows. Taking additionally into account $\left( {\phi _{\mathord {\diamond }}};{\psi _{\mathord {\diamond }}}\right) \in \mathcal{P}\mathcal{R}_{q}\left( {\mathbb {C}\backslash \mathbb {R}}\right)$, we thus obtain $\left( {\phi _{\mathord {\diamond }}};{\psi _{\mathord {\diamond }}}\right) \in \hat{\mathcal {P}}\left[ {M} \right]$, by virtue of Notation 4.10. $\square$

5 $\mathcal {H}$-parameters and the $\mathbb {R}$-quadruple of Matrix Polynomials

In this section, we recall a parametrization of sequences of complex matrices which are related to block Hankel matrices and consider a system of matrix polynomials, which has been proved to be useful in the context of matrix versions of classical moment problems (see, e. g. [1, 2, 5, 7]). Let $(s_j)_{j=0}^{\kappa }$ be a sequence of complex ${p\times q}$ matrices. For every choice of integers $\ell$ and m fulfilling $0\le \ell \le m \le \kappa$, let $y_{\ell ,m}:=\left[ \begin{array}{c}s_{\ell }\\ s_{\ell +1}\\ \vdots \\ s_{m}\end{array}\right]$ and let $z_{\ell ,m}:=\left[ {s_{\ell },s_{\ell +1},\dotsc ,s_{m}} \right]$. Let

$$\begin{aligned} \Theta _{0} :=O_{{p\times q}}\end{aligned}$$

(5.1)

and $\Theta _{n}:=z_{n,2n-1}H_{n-1}^{\mathord {+}}y_{n,2n-1}$ for each $n\in \mathbb {N}$ such that $2n-1\le \kappa$. For each $n\in \mathbb {N}_0$ satisfying $2n\le \kappa$, furthermore, let

$$\begin{aligned} L_{n} :=s_{2n}-\Theta _{n}. \end{aligned}$$

(5.2)

For each $n\in \mathbb {N}$ fulfilling $2n\le \kappa$, let $M_n:=z_{n,2n-1}H_{n-1}^{\mathord {+}}y_{n+1,2n}$ and $N_n:=z_{n+1,2n}H_{n-1}^{\mathord {+}}y_{n,2n-1}$. For all $n\in \mathbb {N}_0$ such that $2n+1\le \kappa$, we also introduce the block Hankel matrix $K_{n}:=\left[ {s_{j+k+1}} \right] _{j,k=0}^{n}$. For every choice of $n\in \mathbb {N}$ fulfilling $2n-1\le \kappa$, we set $\Sigma _n:=z_{n,2n-1}H_{n-1}^{\mathord {+}}K_{n-1}H_{n-1}^{\mathord {+}}y_{n,2n-1}$. Let

$$\begin{aligned} \Lambda _{0} :=O_{{p\times q}}\end{aligned}$$

(5.3)

and $\Lambda _{n}:=M_n+N_n -\Sigma _n$ for all $n\in \mathbb {N}$ fulfilling $2n\le \kappa$.

Remark 5.1

([6, Rem. 5.1]) If $(s_j)_{j=0}^{\kappa }\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,\kappa }$, then $s_{j}^*=s_{j}$ for each $j\in \mathbb {Z}_{0,\kappa }$.

Now we recall the notion of the $\mathcal {H}$-parameter sequence which has been proved to be useful (see, e. g. [4, 5, 7, 16]).

Definition 5.2

([16, Def. 2.3], [5, Def. 5.5]) Let $\kappa \in \mathbb {N}_0\cup \left\{ {\infty } \right\}$ and let $(s_j)_{j=0}^{\kappa }$ be a sequence of complex ${p\times q}$ matrices. The sequence $(\mathfrak {h}_{j})_{j=0}^{\kappa }$ defined by $\mathfrak {h}_{2k}:=s_{2k}-\Theta _{k}$ for each $k\in \mathbb {N}_0$ fulfilling $2k\le \kappa$, and $\mathfrak {h}_{2k+1}:=s_{2k+1}-\Lambda _{k}$ for each $k\in \mathbb {N}_0$ fulfilling $2k+1\le \kappa$, is called the $\mathcal {H}$-parameter sequence of $(s_j)_{j=0}^{\kappa }$.

In view of (5.1) and (5.3), we have in particular $\mathfrak {h}_{0}=s_{0}$ and $\mathfrak {h}_{1}=s_{1}$.

Remark 5.3

(cf. [7, Propositions 2.10(c) and 2.15(b)] and [6, Rem. 6.21]) Let $(s_j)_{j=0}^{\kappa }\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,\kappa }$. Then $\mathfrak {h}_{j}^*=\mathfrak {h}_{j}$ for all $j\in \mathbb {Z}_{0,\kappa }$ and $\mathfrak {h}_{2k}\in \mathbb {C}_\succcurlyeq ^{{q\times q}}$ for all $k\in \mathbb {N}_0$ with $2k\le \kappa$. Furthermore,

$$\begin{aligned}{} & {} \mathfrak {h}_{2k-2}\mathfrak {h}_{2k-2}^{\mathord {+}}\mathfrak {h}_{2k-1}=\mathfrak {h}_{2k-1} \quad \text { and }\quad \mathfrak {h}_{2k-1}\mathfrak {h}_{2k-2}^{\mathord {+}}\mathfrak {h}_{2k-2}=\mathfrak {h}_{2k-1}\nonumber \\{} & {} \quad \text {for all }k\in \mathbb {N}\text { with }2k-1\le \kappa \end{aligned}$$

(5.4)

as well as

$$\begin{aligned}{} & {} \mathfrak {h}_{2k-2}\mathfrak {h}_{2k-2}^{\mathord {+}}\mathfrak {h}_{2k}=\mathfrak {h}_{2k} \text { and } \mathfrak {h}_{2k}\mathfrak {h}_{2k-2}^{\mathord {+}}\mathfrak {h}_{2k-2}=\mathfrak {h}_{2k}\nonumber \\{} & {} \quad \text {for all }k\in \mathbb {N}\text { with }2k\le \kappa . \end{aligned}$$

(5.5)

For each $\tau \in \mathbb {N}_0\cup \left\{ {\infty } \right\}$, let

$$\begin{aligned} \dot{\tau }&:=\sup \left\{ {k\in \mathbb {N}_0}:{2k-1\le \tau }\right\}{} & {} \text {and}&\ddot{\tau }&:=\sup \left\{ {k\in \mathbb {N}_0}:{2k\le \tau }\right\} . \end{aligned}$$

Now we turn our attention to a system of matrix polynomials which plays an essential role in [6, 7].

Definition 5.4

Let $(s_j)_{j=0}^{\kappa }$ be a sequence of complex ${q\times q}$ matrices. Let $\mathfrak {a}_{0}, \mathfrak {b}_{0}, \mathfrak {c}_{0}, \mathfrak {d}_{0}: \mathbb {C}\rightarrow \mathbb {C}^{{q\times q}}$ be defined by

$$\begin{aligned} \mathfrak {a}_{0}\left( {z}\right)&:=O_{{q\times q}},&\mathfrak {b}_{0}\left( {z}\right)&:=I_{q},&\mathfrak {c}_{0}\left( {z}\right)&:=O_{{q\times q}},{} & {} \text {and}&\mathfrak {d}_{0}\left( {z}\right)&:=I_{q}. \end{aligned}$$

(5.6)

If $\kappa \ge 1$, then let $\mathfrak {a}_{1}, \mathfrak {b}_{1}, \mathfrak {c}_{1}, \mathfrak {d}_{1}:\mathbb {C}\rightarrow \mathbb {C}^{{q\times q}}$ be given via

$$\begin{aligned} \mathfrak {a}_{1}\left( {z}\right)&:=\mathfrak {h}_{0},&\mathfrak {b}_{1}\left( {z}\right)&:=zI_{q}-\mathfrak {h}_{0}^{\mathord {+}}\mathfrak {h}_{1},&\mathfrak {c}_{1}\left( {z}\right)&:=\mathfrak {h}_{0},{} & {} \text {and}&\mathfrak {d}_{1}\left( {z}\right)&:=zI_{q}-\mathfrak {h}_{1}\mathfrak {h}_{0}^{\mathord {+}}. \end{aligned}$$

(5.7)

If $\kappa \ge 2$, then, for all $k\in \mathbb {Z}_{2,\infty }$ fulfilling $2k-1\le \kappa$, let $\mathfrak {a}_{k}, \mathfrak {b}_{k}, \mathfrak {c}_{k}, \mathfrak {d}_{k}:\mathbb {C}\rightarrow \mathbb {C}^{{q\times q}}$ be defined recursively by

$$\begin{aligned} \mathfrak {a}_{k}\left( {z}\right)&:=\mathfrak {a}_{k-1}\left( {z}\right) (zI_{q}-\mathfrak {h}_{2k-2}^{\mathord {+}}\mathfrak {h}_{2k-1})-\mathfrak {a}_{k-2}\left( {z}\right) \mathfrak {h}_{2k-4}^{\mathord {+}}\mathfrak {h}_{2k-2},\\ \mathfrak {b}_{k}\left( {z}\right)&:=\mathfrak {b}_{k-1}\left( {z}\right) (zI_{q}-\mathfrak {h}_{2k-2}^{\mathord {+}}\mathfrak {h}_{2k-1})-\mathfrak {b}_{k-2}\left( {z}\right) \mathfrak {h}_{2k-4}^{\mathord {+}}\mathfrak {h}_{2k-2},\\ \mathfrak {c}_{k}\left( {z}\right)&:=\left( {zI_{q}-\mathfrak {h}_{2k-1}\mathfrak {h}_{2k-2}^{\mathord {+}}}\right) \mathfrak {c}_{k-1}\left( {z}\right) -\mathfrak {h}_{2k-2}\mathfrak {h}_{2k-4}^{\mathord {+}}\mathfrak {c}_{k-2}\left( {z}\right) , \end{aligned}$$

and

$$\begin{aligned} \mathfrak {d}_{k}\left( {z}\right)&:=\left( {zI_{q}-\mathfrak {h}_{2k-1}\mathfrak {h}_{2k-2}^{\mathord {+}}}\right) \mathfrak {d}_{k-1}\left( {z}\right) -\mathfrak {h}_{2k-2}\mathfrak {h}_{2k-4}^{\mathord {+}}\mathfrak {d}_{k-2}\left( {z}\right) . \end{aligned}$$

Then we call the quadruple $\left[ {(\mathfrak {a}_{k})_{k=0}^{\dot{\kappa }},(\mathfrak {b}_{k})_{k=0}^{\dot{\kappa }},(\mathfrak {c}_{k})_{k=0}^{\dot{\kappa }},(\mathfrak {d}_{k})_{k=0}^{\dot{\kappa }}} \right]$ the $\mathbb {R}$-quadruple (or canonical quadruple) of matrix polynomials (abbreviating $\mathbb {R}$-QMP) associated with $(s_j)_{j=0}^{\kappa }$.

Notation 5.5

Let $\epsilon :\mathbb {C}\rightarrow \mathbb {C}$ be defined by $\epsilon \left( {z}\right) :=z$.

Remark 5.6

(cf. [6, Rem. 6.14]) Suppose $\kappa \ge 1$. Let $(s_j)_{j=0}^{\kappa }$ be a sequence of complex ${q\times q}$ matrices. For each $k\in \mathbb {N}$ fulfilling $2k+1\le \kappa$, then

and

Observe that $(\mathfrak {b}_{k})_{k=0}^{\kappa }$ is proved to be a monic right orthogonal system of matrix polynomials with respect to $(s_j)_{j=0}^{2\kappa }\in \mathcal {H}^\succcurlyeq _{q,2\kappa }$ (see [7, Thm. 5.5(a)]).

Remark 5.7

(cf. [6, Remarks 6.9 and 7.1]) Let $(s_j)_{j=0}^{\kappa }$ be a sequence of complex ${q\times q}$ matrices and let $k\in \mathbb {N}_0$ be such that $2k-1\le \kappa$. Then $\mathfrak {b}_{k}$ and $\mathfrak {d}_{k}$ are matrix polynomials with degree k and leading coefficient matrix $I_{q}$. In particular, the functions $\det \mathfrak {b}_{k}$ and $\det \mathfrak {d}_{k}$ are polynomials which are not identically vanishing.

Remark 5.8

([6, Rem. 6.15]) Let $(s_j)_{j=0}^{\kappa }$ be a sequence of Hermitian complex ${q\times q}$ matrices. Then $\mathfrak {c}_{k}\left( {z}\right) =\left[ {\mathfrak {a}_{k}\left( {\overline{z}}\right) } \right] ^*$ and $\mathfrak {d}_{k}\left( {z}\right) =\left[ {\mathfrak {b}_{k}\left( {\overline{z}}\right) } \right] ^*$ hold true for every choice of $z\in \mathbb {C}$ and $k\in \mathbb {N}_0$ fulfilling $2k-1\le \kappa$.

Lemma 5.9

(cf. [6, Lem. 6.19]) Let $(s_j)_{j=0}^{\kappa }\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,\kappa }$ and let $k\in \mathbb {N}_0$ be such that $2k-1\le \kappa$. Then $\mathcal {Z}\left( {\det \mathfrak {d}_{k}}\right) =\mathcal {Z}\left( {\det \mathfrak {b}_{k}}\right) \subseteq \mathbb {R}$.

In [4, 6, 7] one can find further results concerning the $\mathbb {R}$-QMP. At the end of this section we introduce a further system of matrix polynomials, which was already considered in [6, Sec. 6].

Notation 5.10

Let $\kappa \in \mathbb {N}_0\cup \left\{ {\infty } \right\}$ and let $(s_j)_{j=0}^{\kappa }$ be a sequence of complex ${q\times q}$ matrices. Then, let $\mathring{\mathfrak {a}}_{1},\mathring{\mathfrak {b}}_{1},\mathring{\mathfrak {c}}_{1},\mathring{\mathfrak {d}}_{1}:\mathbb {C}\rightarrow \mathbb {C}^{{q\times q}}$ be defined by

$$\begin{aligned} \mathring{\mathfrak {a}}_{1}\left( {z}\right)&:=\mathfrak {h}_{0},&\mathring{\mathfrak {b}}_{1}\left( {z}\right)&:=zI_{q},&\mathring{\mathfrak {c}}_{1}\left( {z}\right)&:=\mathfrak {h}_{0},&\mathring{\mathfrak {d}}_{1}\left( {z}\right)&:=zI_{q}. \end{aligned}$$

(5.8)

Furthermore, for all $k\in \mathbb {Z}_{2,\infty }$ fulfilling $2k-2\le \kappa$, let $\mathring{\mathfrak {a}}_{k},\mathring{\mathfrak {b}}_{k},\mathring{\mathfrak {c}}_{k},\mathring{\mathfrak {d}}_{k}:\mathbb {C}\rightarrow \mathbb {C}^{{q\times q}}$ be defined by $\mathring{\mathfrak {a}}_{k}\left( {z}\right) :=z\mathfrak {a}_{k-1}\left( {z}\right) -\mathfrak {a}_{k-2}\left( {z}\right) \mathfrak {h}_{2k-4}^{\mathord {+}}\mathfrak {h}_{2k-2}$ and $\mathring{\mathfrak {b}}_{k}\left( {z}\right) :=z\mathfrak {b}_{k-1}\left( {z}\right) -\mathfrak {b}_{k-2}\left( {z}\right) \mathfrak {h}_{2k-4}^{\mathord {+}}\mathfrak {h}_{2k-2}$ as well as $\mathring{\mathfrak {c}}_{k}\left( {z}\right) :=z\mathfrak {c}_{k-1}\left( {z}\right) -\mathfrak {h}_{2k-2}\mathfrak {h}_{2k-4}^{\mathord {+}}\mathfrak {c}_{k-2}\left( {z}\right)$ and $\mathring{\mathfrak {d}}_{k}\left( {z}\right) :=z\mathfrak {d}_{k-1}\left( {z}\right) -\mathfrak {h}_{2k-2}\mathfrak {h}_{2k-4}^{\mathord {+}}\mathfrak {d}_{k-2}\left( {z}\right)$.

6 Weyl Matrix Balls of a Truncated Hamburger Moment Problem

In [6], the Weyl matrix balls in the context of the matricial versions of the truncated Hamburger moment problem are studied and parametrized without additional assumptions (see Theorem 6.8 below). Some arguments applied there are also useful for our further considerations. First we consider a sequence of rational matrix-valued functions which play an essential role in [6]. In view of Remark 5.7, we recall the corresponding notion.

Definition 6.1

([6, Def. 7.2]) Let $\kappa \in \mathbb {N}_0\cup \left\{ {\infty } \right\}$ and let $(s_j)_{j=0}^{\kappa }\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,\kappa }$. Let $\chi _{-1}:\mathbb {C}\rightarrow \mathbb {C}^{{q\times q}}$ be defined by $\chi _{-1}\left( {z}\right) :=O_{{q\times q}}$. For all $n\in \mathbb {N}_0$ such that $2n\le \kappa$, let $\chi _{2n}:=\mathfrak {h}_{2n}\mathfrak {b}_{n}^{-1}\mathring{\mathfrak {b}}_{n+1}$. For all $n\in \mathbb {N}_0$ fulfilling $2n+1\le \kappa$, let $\chi _{2n+1}:=\mathfrak {h}_{2n}\mathfrak {b}_{n}^{-1}\mathfrak {b}_{n+1}$. Then $(\chi _{j})_{j=-1}^{\kappa }$ is called the sequence of $\chi$-functions associated with $(s_j)_{j=0}^{\kappa }$.

In view of (5.6), (5.8), $\mathfrak {h}_{0}=s_{0}$, (5.7), (5.4), and $\mathfrak {h}_{1}=s_{1}$, for all $z\in \mathbb {C}$, we have

$$\begin{aligned} \chi _{-1}\left( {z}\right)&=O_{{q\times q}},&\chi _{0}\left( {z}\right)&=z\mathfrak {h}_{0}=zs_{0},&\chi _{1}\left( {z}\right)&=z\mathfrak {h}_{0}-\mathfrak {h}_{1}=zs_{0}-s_{1}. \end{aligned}$$

(6.1)

Remark 6.2

(cf. [6, Rem. 7.5(b) and Prop. 7.7(b)]) Let $(s_j)_{j=0}^{\kappa }\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,\kappa }$. For each $k\in \mathbb {N}_0$ such that $2k+1\le \kappa$ and all $z\in \mathbb {C}\backslash \mathbb {R}$, then $\det \mathfrak {b}_{k}\left( {z}\right) \ne 0$ and $\chi _{2k+1}\left( {z}\right) =\mathfrak {h}_{2k}\left[ {\mathfrak {b}_{k}\left( {z}\right) } \right] ^{-1}\mathfrak {b}_{k+1}\left( {z}\right)$ as well as $\det \mathfrak {d}_{k}\left( {z}\right) \ne 0$ and $\chi _{2k+1}\left( {z}\right) =\mathfrak {d}_{k+1}\left( {z}\right) \left[ {\mathfrak {d}_{k}\left( {z}\right) } \right] ^{-1}\mathfrak {h}_{2k}$.

Remark 6.3

([6, Cor. 7.18]) Let $(s_j)_{j=0}^{\kappa }\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,\kappa }$ and let $m\in \mathbb {Z}_{-1,\kappa }$. For all $z\in \mathbb {C}\backslash \mathbb {R}$, then $\left( {\Im z}\right) ^{-1}\Im \chi _{m}\left( {z}\right) \in \mathbb {C}_\succcurlyeq ^{{q\times q}}$.

For each $m\in \mathbb {N}_0$, let

$$\begin{aligned} \dddot{m} :=2\ddot{m} \end{aligned}$$

i. e., if $m=2n$ or $m=2n+1$ for some $n\in \mathbb {N}_0$, then $\ddot{m}=n$ and $\dddot{m}=2n$.

Proposition 6.4

([6, Prop. 7.19]) Let $(s_j)_{j=0}^{\kappa }\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,\kappa }$ and let $m\in \mathbb {Z}_{0,\kappa }$. For all $z\in \mathbb {C}\backslash \mathbb {R}$, then $\mathcal {R}\left( {\left[ {\chi _{m}\left( {z}\right) } \right] ^*}\right) =\mathcal {R}\left( {\chi _{m}\left( {z}\right) }\right) =\mathcal {R}\left( {\Im \chi _{m}\left( {z}\right) }\right) =\mathcal {R}\left( {\mathfrak {h}_{\dddot{m}}}\right)$ and $\mathcal {N}\left( {\left[ {\chi _{m}\left( {z}\right) } \right] ^*}\right) =\mathcal {N}\left( {\chi _{m}\left( {z}\right) }\right) =\mathcal {N}\left( {\Im \chi _{m}\left( {z}\right) }\right) =\mathcal {N}\left( {\mathfrak {h}_{\dddot{m}}}\right)$.

The following statement is a sharpening of Remark 6.3.

Lemma 6.5

Suppose $\kappa \ge 1$. Let $(s_j)_{j=0}^{\kappa }\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,\kappa }$ and let $z\in \mathbb {C}\backslash \mathbb {R}$. For each $k\in \mathbb {N}_0$ fulfilling $2k+1\le \kappa$, then

$$\begin{aligned} \left( {\Im z}\right) ^{-1}\Im \chi _{2k+1}\left( {z}\right) \succcurlyeq \mathfrak {h}_{2k} \succcurlyeq O_{{q\times q}}. \end{aligned}$$

(6.2)

Proof

First observe that Remark 5.3 yields $\mathfrak {h}_{j}^*=\mathfrak {h}_{j}$ for all $j\in \mathbb {Z}_{0,\kappa }$ and $\mathfrak {h}_{2k}\in \mathbb {C}_\succcurlyeq ^{{q\times q}}$ for all $k\in \mathbb {N}_0$ fulfilling $2k\le \kappa$. Since from (6.1) we receive $\chi _{1}\left( {z}\right) =z\mathfrak {h}_{0}-\mathfrak {h}_{1}$, then $\left( {\Im z}\right) ^{-1}\Im \chi _{1}\left( {z}\right) =\mathfrak {h}_{0}\succcurlyeq O_{{q\times q}}$. Now suppose $\kappa \ge 3$. Let $k \in \mathbb {N}$ be such that $2 k +1\le \kappa$. From [6, Cor. 7.22] then we get

$$\begin{aligned}{} & {} \frac{1}{\Im z}\Im \chi _{2k+1}\left( {z}\right) -\mathfrak {h}_{2 k }\\{} & {} \quad =\mathfrak {h}_{2 k }\left[ {\mathfrak {b}_{k}\left( {\overline{z}}\right) } \right] ^{-1}\mathfrak {b}_{k-1}\left( {\overline{z}}\right) \mathfrak {h}_{2k-2}^{\mathord {+}}\left[ {\frac{1}{\Im z}\Im \chi _{2k-1}\left( {z}\right) } \right] \mathfrak {h}_{2k-2}^{\mathord {+}}\mathfrak {d}_{k-1}\left( {z}\right) \left[ {\mathfrak {d}_{k}\left( {z}\right) } \right] ^{-1}\mathfrak {h}_{2 k }. \end{aligned}$$

Because of Remark 5.1, we can infer from Remark 5.8 furthermore $\left[ {\mathfrak {d}_{k}\left( {z}\right) } \right] ^*=\mathfrak {b}_{k}\left( {\overline{z}}\right)$ and $\left[ {\mathfrak {d}_{k-1}\left( {z}\right) } \right] ^*=\mathfrak {b}_{k-1}\left( {\overline{z}}\right)$. Regarding additionally $\mathfrak {h}_{2 k }^*=\mathfrak {h}_{2 k }$ and $\mathfrak {h}_{2k-2}^*=\mathfrak {h}_{2k-2}$ and using Remark A.12, we conclude $\left( {\mathfrak {h}_{2k-2}^{\mathord {+}}\mathfrak {d}_{k-1}\left( {z}\right) \left[ {\mathfrak {d}_{k}\left( {z}\right) } \right] ^{-1}\mathfrak {h}_{2 k }}\right) ^*=\mathfrak {h}_{2 k }\left[ {\mathfrak {b}_{k}\left( {\overline{z}}\right) } \right] ^{-1}\mathfrak {b}_{k-1}\left( {\overline{z}}\right) \mathfrak {h}_{2k-2}^{\mathord {+}}$. Since Remark 6.3 provides $\left( {\Im z}\right) ^{-1}\Im \chi _{2k+1}\left( {z}\right) \in \mathbb {C}_\succcurlyeq ^{{q\times q}}$, we can then apply Remark A.6 to obtain $\left( {\Im z}\right) ^{-1}\Im \chi _{2k+1}\left( {z}\right) -\mathfrak {h}_{2 k }\in \mathbb {C}_\succcurlyeq ^{{q\times q}}$. In view of $\mathfrak {h}_{2 k }\in \mathbb {C}_\succcurlyeq ^{{q\times q}}$, then (6.2) follows. $\square$

Notation 6.6

Let $\kappa \in \mathbb {N}_0\cup \left\{ {\infty } \right\}$ and let $(s_j)_{j=0}^{\kappa }\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,\kappa }$.

• (a) For each $n\in \mathbb {N}_0$ such that $2n\le \kappa$, let $\mathscr {A}_{2n},\mathscr {B}_{2n},\mathscr {C}_{2n}:\mathbb {C}\backslash \mathbb {R}\rightarrow \mathbb {C}^{{q\times q}}$ be defined by

  $$\begin{aligned} \mathscr {A}_{2n}\left( {z}\right)&:=\left[ {\mathfrak {d}_{n}\left( {z}\right) } \right] ^{-1}\mathfrak {h}_{2n}\sqrt{\left( {\Im z}\right) ^{-1}\Im \chi _{2n}\left( {z}\right) }^{\mathord {+}},\\ \mathscr {B}_{2n}\left( {z}\right)&:=\sqrt{\left( {\Im z}\right) ^{-1}\Im \chi _{2n}\left( {z}\right) }^{\mathord {+}}\mathfrak {h}_{2n}\left[ {\mathfrak {b}_{n}\left( {z}\right) } \right] ^{-1}, \end{aligned}$$

  and

  $$\begin{aligned} \mathscr {C}_{2n}\left( {z}\right) :=-\left( {\left[ {\chi _{2n}\left( {z}\right) } \right] ^*\mathfrak {h}_{2n}^{\mathord {+}}\mathfrak {d}_{n}\left( {z}\right) -\mathring{\mathfrak {d}}_{n+1}\left( {z}\right) }\right) ^{-1}\left( {\left[ {\chi _{2n}\left( {z}\right) } \right] ^*\mathfrak {h}_{2n}^{\mathord {+}}\mathfrak {c}_{n}\left( {z}\right) -\mathring{\mathfrak {c}}_{n+1}\left( {z}\right) }\right) . \end{aligned}$$

• (b) For each $n\in \mathbb {N}_0$ such that $2n+1\le \kappa$, let $\mathscr {A}_{2n+1},\mathscr {B}_{2n+1},\mathscr {C}_{2n+1}:\mathbb {C}\backslash \mathbb {R}\rightarrow \mathbb {C}^{{q\times q}}$ be given by

  $$\begin{aligned} \mathscr {A}_{2n+1}\left( {z}\right)&:=\left[ {\mathfrak {d}_{n}\left( {z}\right) } \right] ^{-1}\mathfrak {h}_{2n}\sqrt{\left( {\Im z}\right) ^{-1}\Im \chi _{2n+1}\left( {z}\right) }^{\mathord {+}},\\ \mathscr {B}_{2n+1}\left( {z}\right)&:=\sqrt{\left( {\Im z}\right) ^{-1}\Im \chi _{2n+1}\left( {z}\right) }^{\mathord {+}}\mathfrak {h}_{2n}\left[ {\mathfrak {b}_{n}\left( {z}\right) } \right] ^{-1}, \end{aligned}$$

  and

  $$\begin{aligned} \mathscr {C}_{2n+1}\left( {z}\right) :=-\left( {\left[ {\chi _{2n+1}\left( {z}\right) } \right] ^*\mathfrak {h}_{2n}^{\mathord {+}}\mathfrak {d}_{n}\left( {z}\right) -\mathfrak {d}_{n+1}\left( {z}\right) }\right) ^{-1}\\ \times \left( {\left[ {\chi _{2n+1}\left( {z}\right) } \right] ^*\mathfrak {h}_{2n}^{\mathord {+}}\mathfrak {c}_{n}\left( {z}\right) -\mathfrak {c}_{n+1}\left( {z}\right) }\right) . \end{aligned}$$

Recall that $\mathbb {K}_{{p\times q}}$ stands for the set of all contractive complex ${p\times q}$ matrices.

Notation 6.7

The set $\mathfrak {K}\left( {M;A,B}\right) :=\left\{ {M+AKB}:{K\in \mathbb {K}_{{p\times q}}}\right\}$ signifies the (closed) matrix ball with center M, left semi-radius A, and right semi-radius B with respect to given matrices $M\in \mathbb {C}^{{p\times q}}$, $A\in \mathbb {C}^{{p\times p}}$, and $B\in \mathbb {C}^{{q\times q}}$.

The theory of matrix balls dates back to Yu. L. Shmul’jan [22], who, moreover, examined the operator case in the context of Hilbert spaces.

The set of all values of the solutions of Problem ${\textsf{IP}[\Pi _{\mathord {+}};(s_j)_{j=0}^{2n},\preccurlyeq ]}$ can be described as follows:

Theorem 6.8

([6, Thm. 8.7]) Let $n\in \mathbb {N}_0$ and let $(s_j)_{j=0}^{2n}\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,2n}$. For all $w\in \Pi _{\mathord {+}}$, then $\left\{ {F\left( {w}\right) }:{F\in {\mathcal {R}_{0,q}\left[ {\Pi _{\mathord {+}};(s_j)_{j=0}^{2n},\preccurlyeq } \right] }}\right\} =\mathfrak {K}\left( {\mathscr {C}_{2n}\left( {w}\right) ;\left( {w-\overline{w}}\right) ^{-1}\mathscr {A}_{2n}\left( {w}\right) ,\mathscr {B}_{2n}\left( {w}\right) }\right)$.

We finish this section with three technical results which are needed in the following.

Lemma 6.9

([6, Lemmata 8.10 and 8.12]) Suppose $\kappa \ge 1$. Let $(s_j)_{j=0}^{\kappa }\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,\kappa }$ and let $n\in \mathbb {N}_0$ be such that $2n+1\le \kappa$. For all $z\in \mathbb {C}\backslash \mathbb {R}$, then $\mathscr {A}_{2n}\left( {z}\right) =\mathscr {A}_{2n+1}\left( {z}\right)$ and $\mathscr {B}_{2n}\left( {z}\right) =\mathscr {B}_{2n+1}\left( {z}\right)$ as well as .

Proposition 6.10

(cf. [6, Prop. 8.14]) Let $(s_j)_{j=0}^{\kappa }\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,\kappa }$ and let $z\in \mathbb {C}\backslash \mathbb {R}$. For all $m\in \mathbb {Z}_{0,\kappa }$, then $\mathcal {N}\left( {\mathscr {A}_{m}\left( {z}\right) }\right) =\mathcal {N}\left( {\mathfrak {h}_{\dddot{m}}}\right)$ and $\mathcal {R}\left( {\mathscr {B}_{m}\left( {z}\right) }\right) =\mathcal {R}\left( {\mathfrak {h}_{\dddot{m}}}\right)$.

Proposition 6.11

[6, Prop. 8.18] Let $n\in \mathbb {N}_0$, let $(s_j)_{j=0}^{2n+1}\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,2n+1}$, let $z\in \mathbb {C}\backslash \mathbb {R}$, and let $P:=\mathbb {P}_{\mathcal {R}\left( {\mathfrak {h}_{2n}}\right) }$ and $Q:=\mathbb {P}_{\mathcal {N}\left( {\mathfrak {h}_{2n}}\right) }$. Then there exist matrices $S,T\in \mathbb {C}^{{q\times q}}$ such that the following three conditions are fulfilled:

• (I) $\left( {\Im z}\right) \Im \left( {T^*S}\right) \in \mathbb {C}_\succcurlyeq ^{{q\times q}}$.

• (II) $PS=S$, $SP=S$, and $TP=T-Q$.

• (III) $\det \left( {\mathfrak {b}_{n}\left( {z}\right) \mathfrak {h}_{2n}^{\mathord {+}}S+\mathfrak {b}_{n+1}\left( {z}\right) T}\right) \ne 0$.

If $S,T\in \mathbb {C}^{{q\times q}}$ are arbitrary matrices such that (I)–(III) are fulfilled, then the matrix $\left( {\Im z}\right) ^{-1}\Im \chi _{2n+1}\left( {z}\right)$ is non-negative Hermitian, the matrix

$$\begin{aligned} C:=\sqrt{\left( {\Im z}\right) ^{-1}\Im \chi _{2n+1}\left( {z}\right) }^{\mathord {+}}\left( {S+\left[ {\chi _{2n+1}\left( {z}\right) } \right] ^*T}\right) \\ \times \left( {S+\chi _{2n+1}\left( {z}\right) T}\right) ^{\mathord {+}}\sqrt{\left( {\Im z}\right) ^{-1}\Im \chi _{2n+1}\left( {z}\right) } \end{aligned}$$

is contractive, and the identity

$$\begin{aligned} -\left[ {\mathfrak {a}_{n}\left( {z}\right) \mathfrak {h}_{2n}^{\mathord {+}}S+\mathfrak {a}_{n+1}\left( {z}\right) T} \right] \left[ {\mathfrak {b}_{n}\left( {z}\right) \mathfrak {h}_{2n}^{\mathord {+}}S+\mathfrak {b}_{n+1}\left( {z}\right) T} \right] ^{-1}\\ =\mathscr {C}_{2n+1}\left( {z}\right) +\left( {z-\overline{z}}\right) ^{-1}\mathscr {A}_{2n+1}\left( {z}\right) C\mathscr {B}_{2n+1}\left( {z}\right) \end{aligned}$$

holds true.

7 A Truncated Matricial Stieltjes Moment Problem

Throughout the rest of this paper, let $\alpha \in \mathbb {R}$ be arbitrarily given.

Notation 7.1

Suppose $\kappa \ge 1$. Let $(s_j)_{j=0}^{\kappa }$ be a sequence of complex ${p\times q}$ matrices. Then let the sequence be given by

$$\begin{aligned} a_{j} :=-\alpha s_{j}+s_{j+1}. \end{aligned}$$

(7.1)

For each matrix $X_k$ built from the sequence $(s_j)_{j=0}^{\kappa }$, we denote (if possible) by $X_{\alpha ,k}$ the corresponding matrix built from the sequence instead of $(s_j)_{j=0}^{\kappa }$.

Let $\mathcal {K}^\succcurlyeq _{q,0,\alpha }:=\mathcal {H}^\succcurlyeq _{q,0}$ and, for all $n\in \mathbb {N}$, let $\mathcal {K}^\succcurlyeq _{q,2n,\alpha }$ be the set of all sequences $(s_j)_{j=0}^{2n}$ of complex ${q\times q}$ matrices for which the block Hankel matrices $H_{n}$ and $-\alpha H_{n-1}+K_{n-1}$ are both non-negative Hermitian, i. e., let

$$\begin{aligned} \mathcal {K}^\succcurlyeq _{q,2n,\alpha } :=\left\{ {(s_j)_{j=0}^{2n}\in \mathcal {H}^\succcurlyeq _{q,2n}}:{(a_j)_{j=0}^{2(n-1)}\in \mathcal {H}^\succcurlyeq _{q,2(n-1)}}\right\} . \end{aligned}$$

(7.2)

Furthermore, for all $n\in \mathbb {N}_0$, let $\mathcal {K}^\succcurlyeq _{q,2n+1,\alpha }$ be the set of all sequences $(s_j)_{j=0}^{2n+1}$ of complex ${q\times q}$ matrices for which the block Hankel matrices $H_{n}$ and $-\alpha H_{n}+K_{n}$ are both non-negative Hermitian. A necessary and sufficient criterion for the solvability of the truncated matricial Stieltjes power moment problem can be formulated now as follows:

Theorem 7.2

([3, Thm. 1.4]) Let $m\in \mathbb {N}_0$ and let $(s_j)_{j=0}^{m}$ be a sequence of complex ${q\times q}$ matrices. Then ${\mathcal {M}^{q}_\succcurlyeq [{[\alpha ,\infty )};(s_j)_{j=0}^{m},\preccurlyeq ]}\ne \emptyset$ if and only if $(s_j)_{j=0}^{m}\in \mathcal {K}^\succcurlyeq _{q,m,\alpha }$.

Let $\mathcal {K}^\succcurlyeq _{q,\infty ,\alpha }$ be the set of all sequences $(s_j)_{j=0}^{\infty }$ of complex ${q\times q}$ matrices such that, for all $m\in \mathbb {N}_0$, the sequence $(s_j)_{j=0}^{m}$ belongs to $\mathcal {K}^\succcurlyeq _{q,m,\alpha }$. For each $m\in \mathbb {N}_0$, let $\mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,m,\alpha }$ be the set of all sequences $(s_j)_{j=0}^{m}$ of complex ${q\times q}$ matrices for which there exists a complex ${q\times q}$ matrix $s_{m+1}$ such that $(s_j)_{j=0}^{m+1}$ belongs to $\mathcal {K}^\succcurlyeq _{q,m+1,\alpha }$. For technical reasons, we set $\mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,\infty ,\alpha }:=\mathcal {K}^\succcurlyeq _{q,\infty ,\alpha }$. For all $n\in \mathbb {N}$, we have

$$\begin{aligned} \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,2n,\alpha } =\left\{ {(s_j)_{j=0}^{2n}\in \mathcal {H}^\succcurlyeq _{q,2n}}:{(a_j)_{j=0}^{2n-1}\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,2n-1}}\right\} \end{aligned}$$

(7.3)

and, for all $n\in \mathbb {N}_0$, moreover,

$$\begin{aligned} \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,2n+1,\alpha } =\left\{ {(s_j)_{j=0}^{2n+1}\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,2n+1}}:{(a_j)_{j=0}^{2n}\in \mathcal {H}^\succcurlyeq _{q,2n}}\right\} . \end{aligned}$$

(7.4)

A sequence $(s_j)_{j=0}^{\kappa }$ of complex ${q\times q}$ matrices is called ${[\alpha ,\infty )}$-non-negative definite (resp., ${[\alpha ,\infty )}$-non-negative definite extendable) if it belongs to $\mathcal {K}^\succcurlyeq _{q,\kappa ,\alpha }$ (resp., $\mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,\kappa ,\alpha }$).

Remark 7.3

(cf. [13, Rem. 3.4]) $\mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,\kappa ,\alpha }\subseteq \mathcal {K}^\succcurlyeq _{q,\kappa ,\alpha }$. Furthermore, if $\kappa \ge 1$ and if $(s_j)_{j=0}^{\kappa }\in \mathcal {K}^\succcurlyeq _{q,\kappa ,\alpha }$, then $(s_j)_{j=0}^{\ell }\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,\ell ,\alpha }$ for each $\ell \in \mathbb {Z}_{0,\kappa -1}$.

Corollary 7.4

([13, Cor. 4.18]) Let $m\in \mathbb {N}_0$ and let $(s_j)_{j=0}^{m}\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,m,\alpha }$. Then there exists a sequence $(s_{j})_{j=m+1}^{\infty }$ of complex ${q\times q}$ matrices such that $(s_j)_{j=0}^{\infty }\in \mathcal {K}^\succcurlyeq _{q,\infty ,\alpha }$.

Lemma 7.5

([3, Lemmata 4.7 and 4.11]) Let $n\in \mathbb {N}_0$. Then:

• (a) If $(s_j)_{j=0}^{2n}\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,2n,\alpha }$, then $(s_j)_{j=0}^{2n}\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,2n}$.

• (b) If $(s_j)_{j=0}^{2n+1}\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,2n+1,\alpha }$, then $(a_j)_{j=0}^{2n}\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,2n}$.

Remark 7.6

Suppose $\kappa \ge 1$. Let $(s_j)_{j=0}^{\kappa }\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,\kappa ,\alpha }$. Then, using (7.4), Lemma 7.5, and (7.3) in the case $\kappa <\infty$ as well as Remark 7.3 and (7.2) in the case $\kappa =\infty$, it is readily checked that $(s_j)_{j=0}^{\kappa }\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,\kappa }$ and hold true.

The class $\mathcal {S}_{q}\left( {\mathbb {C}\backslash {[\alpha ,\infty )}}\right)$ of ${q\times q}$ Stieltjes functions in $\mathbb {C}\backslash {[\alpha ,\infty )}$ consists of all holomorphic matrix-valued functions $F:\mathbb {C}\backslash {[\alpha ,\infty )}\rightarrow \mathbb {C}^{{q\times q}}$ satisfying $\Im F\left( {z}\right) \in \mathbb {C}_\succcurlyeq ^{{q\times q}}$ for all $z\in \Pi _{\mathord {+}}$ as well as $F(x)\in \mathbb {C}_\succcurlyeq ^{{q\times q}}$ for all $x\in (-\infty ,\alpha )$. Denote by $\mathcal {S}_{0,q}\left( {\mathbb {C}\backslash {[\alpha ,\infty )}}\right)$ the set of all $F\in \mathcal {S}_{q}\left( {\mathbb {C}\backslash {[\alpha ,\infty )}}\right)$ fulfilling $\sup _{y\in [1,\infty )} y\Vert {F\left( {\textrm{i}y}\right) } \Vert <\infty$. The functions belonging to the class $\mathcal {S}_{0,q}\left( {\mathbb {C}\backslash {[\alpha ,\infty )}}\right)$ admit the following integral representation:

Theorem 7.7

([11, Thm. 5.1])

• (a) For each $F\in \mathcal {S}_{0,q}\left( {\mathbb {C}\backslash {[\alpha ,\infty )}}\right)$, there exists a unique $\sigma \in {\mathcal {M}_\succcurlyeq ^{q}({[\alpha ,\infty )})}$ such that

  $$\begin{aligned} F\left( {z}\right) =\int _{[\alpha ,\infty )}\frac{1}{x -z}\sigma \left( {\textrm{d}x}\right) \quad \text {for all }z\in \mathbb {C}\backslash {[\alpha ,\infty )}. \end{aligned}$$

  (7.5)

• (b) If $\sigma \in {\mathcal {M}_\succcurlyeq ^{q}({[\alpha ,\infty )})}$, then $F:\mathbb {C}\backslash {[\alpha ,\infty )}\rightarrow \mathbb {C}^{{q\times q}}$ defined by (7.5) belongs to $\mathcal {S}_{0,q}\left( {\mathbb {C}\backslash {[\alpha ,\infty )}}\right)$.

If $F\in \mathcal {S}_{0,q}\left( {\mathbb {C}\backslash {[\alpha ,\infty )}}\right)$, then the unique $\sigma \in {\mathcal {M}_\succcurlyeq ^{q}({[\alpha ,\infty )})}$ fulfilling (7.5) is said to be the ${[\alpha ,\infty )}$-spectral measure of F. If $\sigma \in {\mathcal {M}_\succcurlyeq ^{q}({[\alpha ,\infty )})}$, then $F:\mathbb {C}\backslash {[\alpha ,\infty )}\rightarrow \mathbb {C}^{{q\times q}}$ defined by (7.5) is said to be the ${[\alpha ,\infty )}$-Stieltjes transform of $\sigma$.

Remark 7.8

In view of Theorem 7.7, we can now reformulate Problem${\textsf{MP}[{[\alpha ,\infty )};(s_j)_{j=0}^{m},\preccurlyeq ]}$ in the language of ${[\alpha ,\infty )}$-Stieltjes transforms:

Problem

${\textsf{IP}[\mathbb {C}\backslash {[\alpha ,\infty )};(s_j)_{j=0}^{m},\preccurlyeq ]}$ Let $m\in \mathbb {N}_0$ and let $(s_j)_{j=0}^{m}$ be a sequence of complex ${q\times q}$ matrices. Parametrize the set ${\mathcal {S}_{0,q}\left[ {\mathbb {C}\backslash {[\alpha ,\infty )};(s_j)_{j=0}^{m},\preccurlyeq } \right] }$ of all matrix-valued functions $F\in \mathcal {S}_{0,q}\left( {\mathbb {C}\backslash {[\alpha ,\infty )}}\right)$ the $[\alpha , \infty )$-spectral measures of which belong to ${\mathcal {M}^{q}_\succcurlyeq [{[\alpha ,\infty )};(s_j)_{j=0}^{m},\preccurlyeq ]}$.

8 Stieltjes Pairs

In this section, further special pairs of meromorphic matrix-valued functions are considered, which appear as parameter functions in the parametrization of Problem ${\textsf{IP}[\mathbb {C}\backslash {[\alpha ,\infty )};(s_j)_{j=0}^{m},\preccurlyeq ]}$. An important new aspect here is the construction of such pairs in Lemma 8.5 below with prescribed value at a given point $w\in \Pi _{\mathord {+}}$.

Definition 8.1

(cf. [15, Def. 7.1]) Let $\phi _{\mathord {\bullet }}$ and $\psi _{\mathord {\bullet }}$ be ${q\times q}$ matrix-valued functions meromorphic in $\mathbb {C}\backslash {[\alpha ,\infty )}$. The pair $\left( {\phi _{\mathord {\bullet }}};{\psi _{\mathord {\bullet }}}\right)$ is called ${q\times q}$ Stieltjes pair in $\mathbb {C}\backslash {[\alpha ,\infty )}$ if there exists a discrete subset $\mathcal {D}_{\mathord {\bullet }}$ of $\mathbb {C}\backslash {[\alpha ,\infty )}$ such that the following four conditions are fulfilled:

• (I) $\phi _{\mathord {\bullet }}$ and $\psi _{\mathord {\bullet }}$ are holomorphic in $\mathbb {C}\backslash \left( {{[\alpha ,\infty )}\cup \mathcal {D}_{\mathord {\bullet }}}\right)$.

• (II) ${{\,\textrm{rank}\,}}\left[ \begin{array}{l}\phi _{\mathord {\bullet }}\left( {z}\right) \\ \psi _{\mathord {\bullet }}\left( {z}\right) \end{array}\right] =q$ for all $z\in \mathbb {C}\backslash \left( {{[\alpha ,\infty )}\cup \mathcal {D}_{\mathord {\bullet }}}\right)$.

• (III) $\frac{1}{\Im z}\left[ \begin{array}{l}\phi _{\mathord {\bullet }}\left( {z}\right) \\ \psi _{\mathord {\bullet }}\left( {z}\right) \end{array}\right] ^*\left( {-\tilde{J}_{q}}\right) \left[ \begin{array}{l}\phi _{\mathord {\bullet }}\left( {z}\right) \\ \psi _{\mathord {\bullet }}\left( {z}\right) \end{array}\right] \succcurlyeq O_{{q\times q}}$ for all $z\in \mathbb {C}\backslash \left( {\mathbb {R}\cup \mathcal {D}_{\mathord {\bullet }}}\right)$.

• (IV) $\frac{1}{\Im z} \left[ \begin{array}{c} \left( {z-\alpha }\right) \phi _{\mathord {\bullet }}\left( {z}\right) \\ \psi _{\mathord {\bullet }}\left( {z}\right) \end{array}\right] ^*\left( {-\tilde{J}_{q}}\right) \left[ \begin{array}{l}\left( {z-\alpha }\right) \phi _{\mathord {\bullet }}\left( {z}\right) \\ \psi _{\mathord {\bullet }}\left( {z}\right) \end{array}\right] \succcurlyeq O_{{q\times q}}$ for all $z\in \mathbb {C}\backslash \left( {\mathbb {R}\cup \mathcal {D}_{\mathord {\bullet }}}\right)$.

We denote the set of all ${q\times q}$ Stieltjes pairs in $\mathbb {C}\backslash {[\alpha ,\infty )}$ by $\mathcal{P}\mathcal{S}_{q}\left( {\mathbb {C}\backslash {[\alpha ,\infty )}}\right)$. Furthermore, let $\mathscr {D}_{\mathord {\bullet }}\left( {\phi _{\mathord {\bullet }},\psi _{\mathord {\bullet }}}\right)$ be the set of all discrete subsets $\mathcal {D}_{\mathord {\bullet }}$ of $\mathbb {C}\backslash {[\alpha ,\infty )}$ for which the conditions (I)–(IV) hold true.

Remark 8.2

• (a) Remark 4.1 shows that condition (III) in Definition 8.1 can be replaced equivalently by the following condition (III’):

  • (III’) $\left( {\Im z}\right) ^{-1}\Im \left( {\left[ {\psi _{\mathord {\bullet }}\left( {z}\right) } \right] ^*\phi _{\mathord {\bullet }}\left( {z}\right) }\right) \in \mathbb {C}_\succcurlyeq ^{{q\times q}}$ for all $z\in \mathbb {C}\backslash \left( {\mathbb {R}\cup \mathcal {D}_{\mathord {\bullet }}}\right)$.

• (b) Remark 4.1 shows that condition (IV) in Definition 8.1 can be replaced equivalently by the following condition (IV’):

  • (IV’) $\left( {\Im z}\right) ^{-1}\Im \left( {\left( {z-\alpha }\right) \left[ {\psi _{\mathord {\bullet }}\left( {z}\right) } \right] ^*\phi _{\mathord {\bullet }}\left( {z}\right) }\right) \in \mathbb {C}_\succcurlyeq ^{{q\times q}}$ for all $z\in \mathbb {C}\backslash \left( {\mathbb {R}\cup \mathcal {D}_{\mathord {\bullet }}}\right)$.

Remark 8.3

(cf. [15, Remarks 7.3 and 7.5 and Def. 7.4]) Let $\left( {\phi _{\mathord {\bullet }}};{\psi _{\mathord {\bullet }}}\right) \in \mathcal{P}\mathcal{S}_{q}\left( {\mathbb {C}\backslash {[\alpha ,\infty )}}\right)$. For each ${q\times q}$ matrix-valued function $g_{\mathord {\bullet }}$ meromorphic in $\mathbb {C}\backslash {[\alpha ,\infty )}$ such that $\det g_{\mathord {\bullet }}$ does not vanish identically, it is readily checked that the pair $\left( {\phi _{\mathord {\bullet }}g_{\mathord {\bullet }}};{\psi _{\mathord {\bullet }}g_{\mathord {\bullet }}}\right)$ belongs to $\mathcal{P}\mathcal{S}_{q}\left( {\mathbb {C}\backslash {[\alpha ,\infty )}}\right)$ as well. Two pairs $\left( {\phi _{{\mathord {\bullet }},1}};{\psi _{{\mathord {\bullet }},1}}\right)$ and $\left( {\phi _{{\mathord {\bullet }},2}};{\psi _{{\mathord {\bullet }},2}}\right)$ belonging to $\mathcal{P}\mathcal{S}_{q}\left( {\mathbb {C}\backslash {[\alpha ,\infty )}}\right)$ are said to be equivalent if there exists a ${q\times q}$ matrix-valued function $g_{\mathord {\bullet }}$ meromorphic in $\mathbb {C}\backslash {[\alpha ,\infty )}$ such that $\det g_{\mathord {\bullet }}$ does not vanish identically, satisfying $\phi _{{\mathord {\bullet }},2}=\phi _{{\mathord {\bullet }},1} g_{\mathord {\bullet }}$ and $\psi _{{\mathord {\bullet }},2}=\psi _{{\mathord {\bullet }},1}g_{\mathord {\bullet }}$. Indeed, it is readily checked that this relation is an equivalence relation on $\mathcal{P}\mathcal{S}_{q}\left( {\mathbb {C}\backslash {[\alpha ,\infty )}}\right)$. For each $\left( {\phi _{\mathord {\bullet }}};{\psi _{\mathord {\bullet }}}\right) \in \mathcal{P}\mathcal{S}_{q}\left( {\mathbb {C}\backslash {[\alpha ,\infty )}}\right)$, we denote by $\langle {\left( {\phi _{\mathord {\bullet }}};{\psi _{\mathord {\bullet }}}\right) } \rangle$ the equivalence class generated by $\left( {\phi _{\mathord {\bullet }}};{\psi _{\mathord {\bullet }}}\right)$. Furthermore, we write $\langle {\mathcal{P}\mathcal{S}_{q}\left( {\mathbb {C}\backslash {[\alpha ,\infty )}}\right) } \rangle$ for the set of all these equivalence classes.

Notation 8.4

([15, Def. 7.13]) If $M\in \mathbb {C}^{{q\times p}}$, then let $\mathcal {P}_{{\mathord {\bullet }},\alpha }\left[ {M} \right]$ be the set of all $\left( {\phi _{\mathord {\bullet }}};{\psi _{\mathord {\bullet }}}\right) \in \mathcal{P}\mathcal{S}_{q}\left( {\mathbb {C}\backslash {[\alpha ,\infty )}}\right)$ satisfying $\mathbb {P}_{\mathcal {R}\left( {M}\right) }\phi _{\mathord {\bullet }}=\phi _{\mathord {\bullet }}$.

Lemma 8.5

Let $S,T\in \mathbb {C}^{{q\times q}}$ be such that $\Im \left( {T^*S}\right) \in \mathbb {C}_\succcurlyeq ^{{q\times q}}$ and let $\gamma \in \mathbb {C}\backslash \mathbb {R}$. Let $\pi ,\rho :\mathbb {C}\rightarrow \mathbb {C}^{{q\times q}}$ be defined by

$$\begin{aligned} \pi \left( {z}\right) :=S\text { and } \rho \left( {z}\right) :=\frac{\gamma -z}{\Im \gamma }HS+T, \text { where } H:=\left( {S^{\mathord {+}}}\right) ^*\left[ {\Im \left( {T^*S}\right) } \right] S^{\mathord {+}}. \end{aligned}$$

(8.1)

Then:

• (a) $\pi$ and $\rho$ are both holomorphic in $\mathbb {C}$ fulfilling $\pi \left( {\gamma }\right) =S$ and $\rho \left( {\gamma }\right) =T$.

• (b) ${{\,\textrm{rank}\,}}\bigl [{\begin{matrix}\pi \left( {z}\right) \\ \rho \left( {z}\right) \end{matrix}}\bigr ]={{\,\textrm{rank}\,}}\bigl [{\begin{matrix}S\\ T\end{matrix}}\bigr ]$ for all $z\in \mathbb {C}$.

• (c) $\left[ {\rho \left( {z}\right) } \right] ^*\pi \left( {w}\right) =\frac{\overline{\gamma }-\overline{z}}{\Im \gamma }\Im \left( {T^*S}\right) +T^*S$ for all $z,w\in \mathbb {C}$.

• (d) $\left[ {\rho \left( {\overline{z}}\right) } \right] ^*\pi \left( {z}\right) =\left[ {\pi \left( {\overline{z}}\right) } \right] ^*\rho \left( {z}\right)$ for all $z\in \mathbb {C}$.

• (e) $\Im \left( {\left[ {\rho \left( {z}\right) } \right] ^*\pi \left( {z}\right) }\right) =\frac{\Im z}{\Im \gamma }\Im \left( {T^*S}\right)$ for all $z\in \mathbb {C}$.

• (f) $\Im \left( {\left( {z-\xi }\right) \left[ {\rho \left( {z}\right) } \right] ^*\pi \left( {z}\right) }\right) =\frac{\Im z}{\Im \gamma }\Im \left( {\left( {\gamma -\xi }\right) T^*S}\right)$ for all $z\in \mathbb {C}$ and all $\xi \in \mathbb {R}$.

Proof

By virtue of (8.1), we see that part (a) is valid and that

$$\begin{aligned} \begin{bmatrix}\pi \left( {z}\right) \\ \rho \left( {z}\right) \end{bmatrix} = \begin{bmatrix} I_{q}&{}O_{{q\times q}}\\ \frac{\gamma -z}{\Im \gamma }H&{}I_{q}\end{bmatrix} \begin{bmatrix}S\\ T\end{bmatrix} \end{aligned}$$

holds true for all $z\in \mathbb {C}$, which implies part (b). Using Remark A.5, we can infer $\mathcal {N}\left( {S}\right) \subseteq \mathcal {N}\left( {T^*S}\right) \subseteq \mathcal {N}\left( {\Im \left( {T^*S}\right) }\right)$. According to Remark A.14(b), hence $\left[ {\Im \left( {T^*S}\right) } \right] S^{\mathord {+}}S=\Im \left( {T^*S}\right)$. Since $\left[ {\Im \left( {T^*S}\right) } \right] ^*=\Im \left( {T^*S}\right)$, then $S^*\left( {S^{\mathord {+}}}\right) ^*\Im \left( {T^*S}\right) =\left( {S^{\mathord {+}}S}\right) ^*\left[ {\Im \left( {T^*S}\right) } \right] ^*=\left( {\left[ {\Im \left( {T^*S}\right) } \right] S^{\mathord {+}}S}\right) ^*=\left[ {\Im \left( {T^*S}\right) } \right] ^*=\Im \left( {T^*S}\right)$ follows. Regarding (8.1), we consequently have $S^*HS=S^*\left( {S^{\mathord {+}}}\right) ^*\left[ {\Im \left( {T^*S}\right) } \right] S^{\mathord {+}}S=\Im \left( {T^*S}\right)$ and, for all $z,w\in \mathbb {C}$, thus

$$\begin{aligned}\begin{aligned} \left[ {\rho \left( {z}\right) } \right] ^*\pi \left( {w}\right)&=\left( {\frac{\gamma -z}{\Im \gamma }HS+T}\right) ^*S=\left( {\frac{\gamma -z}{\Im \gamma }S^*HS+S^*T}\right) ^*\\&=\left[ {\frac{\gamma -z}{\Im \gamma }\Im \left( {T^*S}\right) +S^*T} \right] ^*=\frac{\overline{\gamma }-\overline{z}}{\Im \gamma }\Im \left( {T^*S}\right) +T^*S. \end{aligned}\end{aligned}$$

Therefore, part (c) is proved. From part (c) we obtain

$$\begin{aligned}\begin{aligned}&\left[ {\rho \left( {\overline{z}}\right) } \right] ^*\pi \left( {z}\right) -\left[ {\pi \left( {\overline{z}}\right) } \right] ^*\rho \left( {z}\right) =\left[ {\rho \left( {\overline{z}}\right) } \right] ^*\pi \left( {z}\right) -\left( {\left[ {\rho \left( {z}\right) } \right] ^*\pi \left( {\overline{z}}\right) }\right) ^*\\&=\left[ {\frac{\overline{\gamma }-z}{\Im \gamma }\Im \left( {T^*S}\right) +T^*S} \right] -\left[ {\frac{\overline{\gamma }-\overline{z}}{\Im \gamma }\Im \left( {T^*S}\right) +T^*S} \right] ^*\\&=\frac{\overline{\gamma }-z}{\Im \gamma }\Im \left( {T^*S}\right) +T^*S-\frac{\gamma -z}{\Im \gamma }\Im \left( {T^*S}\right) -\left( {T^*S}\right) ^*\\&=\frac{\overline{\gamma }-\gamma }{\Im \gamma }\Im \left( {T^*S}\right) +T^*S-\left( {T^*S}\right) ^*=O_{{q\times q}}\end{aligned}\end{aligned}$$

for all $z\in \mathbb {C}$, which implies part (d). Using part (c) and Remarks A.1 and A.3, we conclude

$$\begin{aligned}\begin{aligned} \Im \left( {\left[ {\rho \left( {z}\right) } \right] ^*\pi \left( {z}\right) }\right)&=\Im \left( {\frac{\overline{\gamma }-\overline{z}}{\Im \gamma }\Im \left( {T^*S}\right) +T^*S}\right) =\Im \left( {\frac{\overline{\gamma }-\overline{z}}{\Im \gamma }\Im \left( {T^*S}\right) }\right) +\Im \left( {T^*S}\right) \\&=\frac{\Im \left( {\overline{\gamma }}\right) }{\Im \gamma }\Im \left( {T^*S}\right) -\frac{\Im \left( {\overline{z}}\right) }{\Im \gamma }\Im \left( {T^*S}\right) +\Im \left( {T^*S}\right) \\&=\left( {-\frac{\Im \gamma }{\Im \gamma }+\frac{\Im z}{\Im \gamma }+ 1}\right) \Im \left( {T^*S}\right) =\frac{\Im z}{\Im \gamma }\Im \left( {T^*S}\right) \end{aligned}\end{aligned}$$

for all $z\in \mathbb {C}$, which proves part (e). Let $z\in \mathbb {C}$ and let $\xi \in \mathbb {R}$. Then $\Re \left( {\gamma -\xi }\right) -\left( {\overline{z}-\xi }\right) =\Re \left( {\gamma }\right) -\overline{z}=\overline{\gamma }-\overline{z}+\textrm{i}\Im \left( {\gamma }\right)$. Taking additionally into account Remark A.4 and part (c), we get

$$\begin{aligned}\begin{aligned}&\Im \left( {\left( {\gamma -\xi }\right) T^*S}\right) -\left( {\overline{z}-\xi }\right) \Im \left( {T^*S}\right) \\&=\left[ {\Re \left( {\gamma -\xi }\right) } \right] \Im \left( {T^*S}\right) +\left[ {\Im \left( {\gamma -\xi }\right) } \right] \Re \left( {T^*S}\right) -\left( {\overline{z}-\xi }\right) \Im \left( {T^*S}\right) \\&=\left[ {\Re \left( {\gamma -\xi }\right) -\left( {\overline{z}-\xi }\right) } \right] \Im \left( {T^*S}\right) +\left[ {\Im \left( {\gamma -\xi }\right) } \right] \Re \left( {T^*S}\right) \\&=\left[ {\overline{\gamma }-\overline{z}+\textrm{i}\Im \left( {\gamma }\right) } \right] \Im \left( {T^*S}\right) +\left( {\Im \gamma }\right) \Re \left( {T^*S}\right) \\&=\left( {\overline{\gamma }-\overline{z}}\right) \Im \left( {T^*S}\right) +\left( {\Im \gamma }\right) \left[ {\textrm{i}\Im \left( {T^*S}\right) +\Re \left( {T^*S}\right) } \right] \\&=\left( {\overline{\gamma }-\overline{z}}\right) \Im \left( {T^*S}\right) +\left( {\Im \gamma }\right) T^*S=\left( {\Im \gamma }\right) \left[ {\rho \left( {z}\right) } \right] ^*\pi \left( {z}\right) . \end{aligned}\end{aligned}$$

Using this and Remark A.1, we obtain

$$\begin{aligned}\begin{aligned}&\left( {\Im \gamma }\right) \Im \left( {\left( {z-\xi }\right) \left[ {\rho \left( {z}\right) } \right] ^*\pi \left( {z}\right) }\right) =\Im \left( {\left( {z-\xi }\right) \left( {\Im \gamma }\right) \left[ {\rho \left( {z}\right) } \right] ^*\pi \left( {z}\right) }\right) \\&=\Im \left( {\left( {z-\xi }\right) \left[ {\Im \left( {\left( {\gamma -\xi }\right) T^*S}\right) -\left( {\overline{z}-\xi }\right) \Im \left( {T^*S}\right) } \right] }\right) \\&=\Im \left( {\left( {z-\xi }\right) \Im \left( {\left( {\gamma -\xi }\right) T^*S}\right) -|{z-\xi } |^2\Im \left( {T^*S}\right) }\right) \\&=\Im \left( {\left( {z-\xi }\right) \Im \left( {\left( {\gamma -\xi }\right) T^*S}\right) }\right) -\Im \left( {|{z-\xi } |^2\Im \left( {T^*S}\right) }\right) \\&=\left[ {\Im \left( {z-\xi }\right) } \right] \Im \left( {\left( {\gamma -\xi }\right) T^*S}\right) =\left( {\Im z}\right) \Im \left( {\left( {\gamma -\xi }\right) T^*S}\right) . \end{aligned}\end{aligned}$$

Therefore, part (f) is proved as well. $\square$

9 $\mathcal {K}_\alpha$-parameters

We recall the concept of $\mathcal {K}_\alpha$-parameter sequences which is taken from [8]. We list some of its properties and add some minor technical aspects.

Definition 9.1

([8, Def. 4.2]) Let $(s_j)_{j=0}^{\kappa }$ be a sequence of complex ${p\times q}$ matrices. The sequence $(\mathfrak {k}_{j})_{j=0}^{\kappa }$ defined by $\mathfrak {k}_{2k}:=L_{k}$ for each $k\in \mathbb {N}_0$ such that $2k\le \kappa$, and $\mathfrak {k}_{2k+1}:=L_{{\alpha ,k}}$ for each $k\in \mathbb {N}_0$ such that $2k+1\le \kappa$, is called the $\mathcal {K}_\alpha$-parameter sequence of $(s_j)_{j=0}^{\kappa }$.

Regarding (5.2) and (5.1), we have in particular $\mathfrak {k}_{0}=s_{0}$ and $\mathfrak {k}_{1}=a_{0}$. If $\kappa \ge 1$ and $(s_j)_{j=0}^{\kappa }$ is a sequence of complex ${p\times q}$ matrices, then, in accordance with Notation 7.1 and Definition 5.2, we denote in the sequel by $(\mathfrak {h}_{{\alpha ,j}})_{j=0}^{\kappa -1}$ the $\mathcal {H}$-parameter sequence of $(a_j)_{j=0}^{\kappa -1}$.

Remark 9.2

([8, Rem. 6.1]) Let $(s_j)_{j=0}^{\kappa }$ be a sequence of complex ${p\times q}$ matrices. Regarding (5.2), then one can easily see from Definitions 9.1 and 5.2 that

$$\begin{aligned} \mathfrak {k}_{2k}=\mathfrak {h}_{2k}\quad \text {for all }k \in \mathbb {N}_0\text { with }2k\le \kappa \end{aligned}$$

(9.1)

and, in the case $\kappa \ge 1$, moreover, that

$$\begin{aligned} \mathfrak {k}_{2k+1}=\mathfrak {h}_{{\alpha ,2k}} \quad \text {for all }k \in \mathbb {N}_0\text { with }2k+1\le \kappa . \end{aligned}$$

(9.2)

Remark 9.3

Let $(s_j)_{j=0}^{\kappa }\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,\kappa ,\alpha }$. From [8, Thm. 4.12] one can see that $s_{j}^*=s_{j}$ and $\mathfrak {k}_{j}^*=\mathfrak {k}_{j}$ for all $j\in \mathbb {Z}_{0,\kappa }$ as well as $\mathcal {N}\left( {\mathfrak {k}_{\ell }}\right) \subseteq \mathcal {N}\left( {\mathfrak {k}_{m}}\right)$ for all $\ell ,m\in \mathbb {Z}_{0,\kappa }$ with $\ell \le m$. Remark A.7 then shows that $\mathcal {R}\left( {\mathfrak {k}_{m}}\right) \subseteq \mathcal {R}\left( {\mathfrak {k}_{\ell }}\right)$ for all $\ell ,m\in \mathbb {Z}_{0,\kappa }$ with $\ell \le m$, which in view of Remark A.14, implies

$$\begin{aligned} \mathfrak {k}_{\ell }\mathfrak {k}_{\ell }^{\mathord {+}}\mathfrak {k}_{ m }&=\mathfrak {k}_{ m }{} & {} \text {and}&\mathfrak {k}_{ m }\mathfrak {k}_{\ell }^{\mathord {+}}\mathfrak {k}_{\ell }&=\mathfrak {k}_{ m }&\text {for all } \ell , m&\in \mathbb {Z}_{0,\kappa }\text { with } \ell \le m . \end{aligned}$$

(9.3)

Consequently, Remark A.13 provides $\mathcal {R}\left( {\mathfrak {k}_{ m }^{\mathord {+}}}\right) \subseteq \mathcal {R}\left( {\mathfrak {k}_{\ell }^{\mathord {+}}}\right)$ and $\mathcal {N}\left( {\mathfrak {k}_{\ell }^{\mathord {+}}}\right) \subseteq \mathcal {N}\left( {\mathfrak {k}_{ m }^{\mathord {+}}}\right)$ for all $\ell , m \in \mathbb {Z}_{0,\kappa }$ with $\ell \le m$ and, because of Remarks A.12 and A.14, then

$$\begin{aligned} \mathfrak {k}_{\ell }^{\mathord {+}}\mathfrak {k}_{\ell }\mathfrak {k}_{ m }^{\mathord {+}}&=\mathfrak {k}_{ m }^{\mathord {+}}{} & {} \text {and}&\mathfrak {k}_{ m }^{\mathord {+}}\mathfrak {k}_{\ell }\mathfrak {k}_{\ell }^{\mathord {+}}&=\mathfrak {k}_{ m }^{\mathord {+}}&\text {for all } \ell , m&\in \mathbb {Z}_{0,\kappa }\text { with } \ell \le m . \end{aligned}$$

(9.4)

Lemma 9.4

(cf. [8, Lem.  6.7 and Thm. 6.8]) Let $(s_j)_{j=0}^{\kappa }\in \mathcal {K}^\succcurlyeq _{q,\kappa ,\alpha }$. For all $k\in \mathbb {N}_0$ fulfilling $2k\le \kappa$, then $\mathfrak {h}_{2k}=\mathfrak {k}_{2k}$. If $\kappa \ge 1$, then

$$\begin{aligned}{} & {} \mathfrak {h}_{1}=\mathfrak {k}_{1}+\alpha \mathfrak {k}_{0} \text { and } \mathfrak {h}_{2k+1}=\mathfrak {k}_{2k+1}+\alpha \mathfrak {k}_{2k}+\mathfrak {k}_{2k}\mathfrak {k}_{2k-1}^{\mathord {+}}\mathfrak {k}_{2k}\nonumber \\{} & {} \quad \text {for all }k\in \mathbb {N}\text { with }2k+1\le \kappa . \end{aligned}$$

(9.5)

Lemma 9.5

Suppose $\kappa \ge 1$. Let $(s_j)_{j=0}^{\kappa }\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,\kappa ,\alpha }$. Then $\mathfrak {h}_{0}\left( {\alpha I_{q}+\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{1}}\right) =\mathfrak {h}_{1}$ and $\left( {\alpha I_{q}+\mathfrak {k}_{1}\mathfrak {k}_{0}^{\mathord {+}}}\right) \mathfrak {h}_{0}=\mathfrak {h}_{1}$. If $\kappa \ge 3$, for all $k \in \mathbb {N}$ fulfilling $2 k +1\le \kappa$, then $\mathfrak {h}_{2 k }\left( {\alpha I_{q}+\mathfrak {k}_{2k-1}^{\mathord {+}}\mathfrak {k}_{2k}+\mathfrak {k}_{2k}^{\mathord {+}}\mathfrak {k}_{2k+1}}\right) =\mathfrak {h}_{2k+1}$ and $\left( {\alpha I_{q}+\mathfrak {k}_{2k}\mathfrak {k}_{2k-1}^{\mathord {+}}+\mathfrak {k}_{2k+1}\mathfrak {k}_{2k}^{\mathord {+}}}\right) \mathfrak {h}_{2 k }=\mathfrak {h}_{2k+1}$.

Proof

First observe that Remark 9.3 yields (9.3), whereas Remark 7.3 provides $(s_j)_{j=0}^{\kappa }\in \mathcal {K}^\succcurlyeq _{q,\kappa ,\alpha }$. Remark 9.2 provides (9.1). Consequently, we can apply (9.3) and Lemma 9.4 to obtain $\mathfrak {h}_{0}\left( {\alpha I_{q}+\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{1}}\right) =\mathfrak {k}_{0}\left( {\alpha I_{q}+\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{1}}\right) =\alpha \mathfrak {k}_{0}+\mathfrak {k}_{0}\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{1}=\alpha \mathfrak {k}_{0}+\mathfrak {k}_{1}=\mathfrak {h}_{1}$ and, analogously, $\left( {\alpha I_{q}+\mathfrak {k}_{1}\mathfrak {k}_{0}^{\mathord {+}}}\right) \mathfrak {h}_{0}=\mathfrak {h}_{1}$. Now suppose $\kappa \ge 3$. Let $k\in \mathbb {N}$ be such that $2k+1\le \kappa$. Using (9.3) and Lemma 9.4, we can infer then

$$\begin{aligned}\begin{aligned}&\mathfrak {h}_{2 k }\left( {\alpha I_{q}+\mathfrak {k}_{2k-1}^{\mathord {+}}\mathfrak {k}_{2k}+\mathfrak {k}_{2k}^{\mathord {+}}\mathfrak {k}_{2k+1}}\right) =\mathfrak {k}_{2 k }\left( {\alpha I_{q}+\mathfrak {k}_{2k-1}^{\mathord {+}}\mathfrak {k}_{2k}+\mathfrak {k}_{2k}^{\mathord {+}}\mathfrak {k}_{2k+1}}\right) \\&=\alpha \mathfrak {k}_{2 k }+\mathfrak {k}_{2 k }\mathfrak {k}_{2k-1}^{\mathord {+}}\mathfrak {k}_{2k}+\mathfrak {k}_{2 k }\mathfrak {k}_{2k}^{\mathord {+}}\mathfrak {k}_{2k+1} =\alpha \mathfrak {k}_{2 k }+\mathfrak {k}_{2 k }\mathfrak {k}_{2k-1}^{\mathord {+}}\mathfrak {k}_{2k}+\mathfrak {k}_{2k+1} =\mathfrak {h}_{2k+1} \end{aligned}\end{aligned}$$

and, analogously, $\left( {\alpha I_{q}+\mathfrak {k}_{2k}\mathfrak {k}_{2k-1}^{\mathord {+}}+\mathfrak {k}_{2k+1}\mathfrak {k}_{2k}^{\mathord {+}}}\right) \mathfrak {h}_{2 k }=\mathfrak {h}_{2k+1}$. $\square$

Lemma 9.6

(cf. Remark 9.2 and [8, Lem. 6.19 and Thm. 6.20]) Suppose $\kappa \ge 1$. Let $(s_j)_{j=0}^{\kappa }\in \mathcal {K}^\succcurlyeq _{q,\kappa ,\alpha }$. For all $k\in \mathbb {N}_0$ fulfilling $2k\le \kappa -1$, then $\mathfrak {h}_{{\alpha ,2k}}=\mathfrak {k}_{2k+1}$. If $\kappa \ge 2$, then

$$\begin{aligned}{} & {} \mathfrak {h}_{{\alpha ,2k+1}}=\mathfrak {k}_{2k+2}+\alpha \mathfrak {k}_{2k+1}+\mathfrak {k}_{2k+1}\mathfrak {k}_{2k}^{\mathord {+}}\mathfrak {k}_{2k+1}\nonumber \\{} & {} \quad \text {for all }k\in \mathbb {N}_0\text { with }2k+1\le \kappa -1. \end{aligned}$$

(9.6)

Lemma 9.7

Suppose $\kappa \ge 2$. Let $(s_j)_{j=0}^{\kappa }\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,\kappa ,\alpha }$ and let $k \in \mathbb {N}_0$ be such that $2 k +2\le \kappa$. Then, $\mathfrak {h}_{{\alpha ,2 k}}\left( {\alpha I_{q}+\mathfrak {k}_{2k}^{\mathord {+}}\mathfrak {k}_{2k+1}+\mathfrak {k}_{2k+1}^{\mathord {+}}\mathfrak {k}_{2k+2}}\right) =\mathfrak {h}_{{\alpha ,2k+1}}$ and $\left( {\alpha I_{q}+\mathfrak {k}_{2k+1}\mathfrak {k}_{2k}^{\mathord {+}}+\mathfrak {k}_{2k+2}\mathfrak {k}_{2k+1}^{\mathord {+}}}\right) \mathfrak {h}_{{\alpha ,2 k }}=\mathfrak {h}_{{\alpha ,2k+1}}$.

Proof

First observe that Remark 9.3 yields (9.3), whereas Remark 7.3 provides $(s_j)_{j=0}^{\kappa }\in \mathcal {K}^\succcurlyeq _{q,\kappa ,\alpha }$. Consequently, we can apply (9.3) and Lemma 9.6 to obtain

$$\begin{aligned}\begin{aligned}&\mathfrak {h}_{{\alpha ,2 k }}\left( {\alpha I_{q}+\mathfrak {k}_{2k}^{\mathord {+}}\mathfrak {k}_{2k+1}+\mathfrak {k}_{2k+1}^{\mathord {+}}\mathfrak {k}_{2k+2}}\right) =\mathfrak {k}_{2 k+1}\left( {\alpha I_{q}+\mathfrak {k}_{2k}^{\mathord {+}}\mathfrak {k}_{2k+1}+\mathfrak {k}_{2k+1}^{\mathord {+}}\mathfrak {k}_{2k+2}}\right) \\&=\alpha \mathfrak {k}_{2 k+1}+\mathfrak {k}_{2 k+1}\mathfrak {k}_{2k}^{\mathord {+}}\mathfrak {k}_{2k+1}+\mathfrak {k}_{2 k+1}\mathfrak {k}_{2k+1}^{\mathord {+}}\mathfrak {k}_{2k+2}\\&=\alpha \mathfrak {k}_{2 k+1}+\mathfrak {k}_{2 k+1}\mathfrak {k}_{2k}^{\mathord {+}}\mathfrak {k}_{2k+1}+\mathfrak {k}_{2k+2} =\mathfrak {h}_{{\alpha ,2k+1}} \end{aligned}\end{aligned}$$

and, analogously, $\left( {\alpha I_{q}+\mathfrak {k}_{2k+1}\mathfrak {k}_{2k}^{\mathord {+}}+\mathfrak {k}_{2k+2}\mathfrak {k}_{2k+1}^{\mathord {+}}}\right) \mathfrak {h}_{{\alpha ,2 k }}=\mathfrak {h}_{{\alpha ,2k+1}}$. $\square$

10 Particular Matrix Polynomials Related to Some Matricial Stieltjes Moment Problem

This section is aimed at presenting technical aspects of certain matrix polynomials. These matrix polynomials generate a parametrization of the solution set of Problem ${\textsf{IP}[\mathbb {C}\backslash {[\alpha ,\infty )};(s_j)_{j=0}^{m},=]}$ in form of a linear fractional transformation, which we will recall at the end of this section.

Given $m\in \mathbb {N}$ and arbitrary rectangular complex matrices $A_1,A_2,\dotsc ,A_m$, we write ${{\,\textrm{col}\,}}(A_j)_{j=1}^{m}$ (resp., ${{\,\textrm{row}\,}}(A_k)_{k=1}^{m}$) for the block column (resp., block row) built from the matrices $A_1,A_2,\dotsc ,A_m$ if their numbers of columns (resp., rows) are all equal.

Notation 10.1

Let P be a complex ${p\times q}$ matrix polynomial. For each $n\in \mathbb {N}_0$, let $Z_{P,n}:={{\,\textrm{row}\,}}(A_k)_{k=0}^{n}$ and $Y_{P,n}:={{\,\textrm{col}\,}}(A_j)_{j=0}^{n}$, where $(A_\ell )_{\ell =0}^{\infty }$ is the uniquely determined sequence of complex ${p\times q}$ matrices, such that $P\left( {z}\right) =\sum _{\ell =0}^\infty z^\ell A_\ell$ holds true for all $z\in \mathbb {C}$. Denote by $\deg P:=\sup \left\{ {\ell \in \mathbb {N}_0}:{A_\ell \ne O_{{p\times q}}}\right\}$ the degree of P. If $m:=\deg P\ge 0$, then the matrix $A_m$ is called the leading coefficient matrix of P.

Remark 10.2

If P is a complex ${q\times q}$ matrix polynomial, then $P=E_{n}Y_{P,n}$ for all $n\in \mathbb {N}_0$ with $n\ge \deg P$, where $E_{n}:\mathbb {C}\rightarrow \mathbb {C}^{{q\times (n+1)q}}$ is defined by $E_{n}(z):=\left[ {z^0I_{q},z^1I_{q},z^2I_{q},\dotsc ,z^nI_{q}} \right]$.

Let $(s_j)_{j=0}^{\kappa }$ be a sequence of complex ${p\times q}$ matrices. For all $m\in \mathbb {Z}_{0,\kappa }$, then let

$$\begin{aligned} \textbf{S}_{m}&:=\left[ \begin{array}{ccccc} s_{0}&{}O&{}O&{}\cdots &{}O\\ s_1&{}s_{0}&{}O&{}\cdots &{}O\\ s_2&{}s_1&{}s_{0}&{}\cdots &{}O\\ \vdots &{}\vdots &{}\vdots &{}\ddots &{}\vdots \\ s_m&{}s_{m-1}&{}s_{m-2}&{}\cdots &{}s_{0} \end{array}\right]{} & {} \text {and}&\mathbb {S}_{m}&:=\left[ \begin{array}{ccccc} s_0 &{} s_1 &{} s_2 &{} \ldots &{} s_m \\ O&{} s_0 &{} s_1 &{} \ldots &{}s_{m-1} \\ O&{} O&{} s_0 &{} \ldots &{}s_{m-2} \\ \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ O&{} O&{} O&{} \ldots &{} s_0 \end{array}\right] . \end{aligned}$$

(10.1)

Notation 10.3

Let $(s_j)_{j=0}^{\kappa }$ be a sequence of complex ${q\times q}$ matrices and let P be a complex ${q\times q}$ matrix polynomial with degree $n:=\deg P$ satisfying $n\le \kappa +1$. Then let $P^{\langle s\rangle },P^{\llbracket s\rrbracket }:\mathbb {C}\rightarrow \mathbb {C}^{{q\times q}}$ be defined by $P^{\langle s\rangle }\left( {z}\right) :=O_{{q\times q}}$ and $P^{\llbracket s\rrbracket }\left( {z}\right) :=O_{{q\times q}}$ if $n\le 0$ and by $P^{\langle s\rangle }\left( {z}\right) :=Z_{P,n}\bigl [{\begin{matrix}O_{{q\times nq}}\\ \textbf{S}_{n-1}\end{matrix}}\bigr ]\left[ {E_{n-1}(\overline{z})} \right] ^*$ and $P^{\llbracket s\rrbracket }\left( {z}\right) :=E_{n-1}(z)\left[ {O_{{nq\times q}},\mathbb {S}_{n-1}} \right] Y_{P,n}$ if $n\ge 1$.

Lemma 10.4

Let $(s_j)_{j=0}^{\kappa }$ be a sequence of Hermitian complex ${q\times q}$ matrices and let P be a complex ${q\times q}$ matrix polynomial such that $\deg P\le \kappa +1$. Then $Q:=P^{\mathord {\vee }}$ is a complex ${q\times q}$ matrix polynomial with $\deg Q=\deg P$ such that $Q^{\langle s\rangle }=\left( { P^{\llbracket s\rrbracket }}\right) ^{\mathord {\vee }}$ is valid, i. e., $Q^{\langle s\rangle }\left( {z}\right) =\left[ { P^{\llbracket s\rrbracket }\left( {\overline{z}}\right) } \right] ^*$ holds true for all $z\in \mathbb {C}$.

Proof

Let $n:=\deg P$. If $n\le 0$, then, in view of (2.2) and Notations 10.1 and 10.3, the assertion is obvious. Now suppose $n\ge 1$. Then there are complex ${q\times q}$ matrices $A_0,A_1,\dotsc ,A_n$ such that $P\left( {z}\right) =\sum _{\ell =0}^nz^\ell A_\ell$ for all $z\in \mathbb {C}$ and $A_n\ne O_{{q\times q}}$. Regarding (2.2), hence $Q\left( {z}\right) =\left[ {P\left( {\overline{z}}\right) } \right] ^*=\sum _{\ell =0}^nz^\ell A_\ell ^*$ for all $z\in \mathbb {C}$ and $A_n^*\ne O_{{q\times q}}$. Consequently, Q is a complex ${q\times q}$ matrix polynomial with $\deg Q=n$. In view of Notation 10.1, we can furthermore infer $Y_{P,n}={{\,\textrm{col}\,}}\left( {A_j}\right) _{j=0}^n$ and $Z_{Q,n}={{\,\textrm{row}\,}}\left( {A_k^*}\right) _{k=0}^n$. Thus, $Y_{P,n}^*=Z_{Q,n}$ follows. Since $(s_j)_{j=0}^{\kappa }$ is a sequence of Hermitian matrices, it is easily seen from (10.1) that $\mathbb {S}_{n-1}^*=\textbf{S}_{n-1}$. According to Notation 10.3 we have $Q^{\langle s\rangle }\left( {z}\right) =Z_{Q,n}\bigl [{\begin{matrix}O_{{q\times nq}}\\ \textbf{S}_{n-1}\end{matrix}}\bigr ]\left[ {E_{n-1}(\overline{z})} \right] ^*$ and $P^{\llbracket s\rrbracket }\left( {\overline{z}}\right) =E_{n-1}(\overline{z})\left[ {O_{{nq\times q}},\mathbb {S}_{n-1}} \right] Y_{P,n}$ for all $z\in \mathbb {C}$. Summarizing, we conclude $\left[ {P^{\llbracket s\rrbracket }\left( {\overline{z}}\right) } \right] ^*=Y_{P,n}^*\bigl [{\begin{matrix}O_{{q\times nq}}\\ \mathbb {S}_{n-1}^*\end{matrix}}\bigr ]\left[ {E_{n-1}(\overline{z})} \right] ^*=Q^{\langle s\rangle }\left( {z}\right)$ for all $z\in \mathbb {C}$. Regarding (2.2), hence $\left( {P^{\llbracket s\rrbracket }}\right) ^{\mathord {\vee }}=Q^{\langle s\rangle }$.

Lemma 10.5

(cf. [13, Lem. E.5]) Let $(s_j)_{j=0}^{\kappa }$ be a sequence of complex ${q\times q}$ matrices and let P be a complex ${q\times q}$ matrix polynomial with degree k satisfying $k\le \kappa$. Let the matrix polynomial Q be given by $Q\left( {z}\right) :=\left( {z-\alpha }\right) P\left( {z}\right)$. Then $Q^{\llbracket s\rrbracket }=s_{0}P$ if $k\le 0$ and $Q^{\llbracket s\rrbracket }=P^{\llbracket a\rrbracket }+s_{0}P$ if $k\ge 1$.

Notation 10.6

For each $n\in \mathbb {N}_0$, let $\varepsilon _{2n},\varepsilon _{2n+1}:\mathbb {C}\rightarrow \mathbb {C}$ be defined by $\varepsilon _{2n}\left( {z}\right) :=z-\alpha$ and $\varepsilon _{2n+1}\left( {z}\right) :=1$, respectively.

Notation 10.7

(cf. [13, Notation 14.1 and Rem. 14.2]) Let $(s_j)_{j=0}^{\kappa }$ be a sequence of complex ${q\times q}$ matrices with $\mathcal {K}_\alpha$-parameter sequence $(\mathfrak {k}_{j})_{j=0}^{\kappa }$. Then let $\textbf{p}_{0},\textbf{q}_{0},\textbf{r}_{0},\textbf{t}_{0},\textbf{p}_{1},\textbf{q}_{1},\textbf{r}_{1},\textbf{t}_{1}:\mathbb {C}\rightarrow \mathbb {C}^{{q\times q}}$ be defined by

$$\begin{aligned} \textbf{p}_{0}\left( {z}\right)&:=I_{q},&\textbf{q}_{0}\left( {z}\right)&:=O_{{q\times q}},&\textbf{r}_{0}\left( {z}\right)&:=I_{q},&\textbf{t}_{0}\left( {z}\right)&:=O_{{q\times q}} \end{aligned}$$

(10.2)

and

$$\begin{aligned} \textbf{p}_{1}\left( {z}\right)&:=\left( {z-\alpha }\right) I_{q},&\textbf{q}_{1}\left( {z}\right)&:=\mathfrak {k}_{0},&\textbf{r}_{1}\left( {z}\right)&:=\left( {z-\alpha }\right) I_{q},&\textbf{t}_{1}\left( {z}\right)&:=\mathfrak {k}_{0}, \end{aligned}$$

(10.3)

respectively. If $\kappa \ge 1$, then, for all $\ell \in \mathbb {Z}_{2,\kappa +1}$, let $\textbf{p}_{\ell },\textbf{q}_{\ell },\textbf{r}_{\ell },\textbf{t}_{\ell }:\mathbb {C}\rightarrow \mathbb {C}^{{q\times q}}$ be given recursively by

$$\begin{aligned} \textbf{p}_{\ell }&:=\varepsilon _{\ell -1}\textbf{p}_{\ell -1}-\textbf{p}_{\ell -2}\mathfrak {k}_{\ell -2}^{\mathord {+}}\mathfrak {k}_{\ell -1},&\textbf{q}_{\ell }&:=\varepsilon _{\ell -1}\textbf{q}_{\ell -1}-\textbf{q}_{\ell -2}\mathfrak {k}_{\ell -2}^{\mathord {+}}\mathfrak {k}_{\ell -1} \end{aligned}$$

(10.4)

and

$$\begin{aligned} \textbf{r}_{\ell }&:=\varepsilon _{\ell -1}\textbf{r}_{\ell -1} -\mathfrak {k}_{\ell -1}\mathfrak {k}_{\ell -2}^{\mathord {+}}\textbf{r}_{\ell -2},&\textbf{t}_{\ell }&:=\varepsilon _{\ell -1}\textbf{t}_{\ell -1} -\mathfrak {k}_{\ell -1}\mathfrak {k}_{\ell -2}^{\mathord {+}}\textbf{t}_{\ell -2}, \end{aligned}$$

(10.5)

respectively.

Obviously, for each $\ell \in \mathbb {Z}_{0,\kappa +1}$, the functions $\textbf{p}_{\ell }$, $\textbf{q}_{\ell }$, $\textbf{r}_{\ell }$, and $\textbf{t}_{\ell }$ are complex ${q\times q}$ matrix polynomials.

Remark 10.8

(cf. [13, Rem. 14.4]) Let $(s_j)_{j=0}^{\kappa }$ be a sequence of complex ${q\times q}$ matrices. For each $k \in \mathbb {N}_0$ fulfilling $2 k -1\le \kappa$, the functions $\textbf{p}_{2 k }$ and $\textbf{r}_{2 k }$ are complex ${q\times q}$ matrix polynomials with degree k and leading coefficient matrix $I_{q}$. For each $k \in \mathbb {N}_0$ fulfilling $2 k \le \kappa$, the functions $\textbf{p}_{2k+1}$ and $\textbf{r}_{2k+1}$ are complex ${q\times q}$ matrix polynomials with degree $k +1$ and leading coefficient matrix $I_{q}$, satisfying $\textbf{p}_{2k+1}\left( {\alpha }\right) =O_{{q\times q}}$ and $\textbf{r}_{2k+1}\left( {\alpha }\right) =O_{{q\times q}}$.

Remark 10.9

Suppose $\kappa \ge 1$. Let $(s_j)_{j=0}^{\kappa }$ be a sequence of complex ${q\times q}$ matrices. In view of Notations 10.7 and 10.6 and Remark 10.8, by mathematical induction one can easily check that $\textbf{p}_{2k}\left( {\alpha }\right) =\left( {-1}\right) ^k\left( {\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{1}}\right) \left( {\mathfrak {k}_{2}^{\mathord {+}}\mathfrak {k}_{3}}\right) \cdots \left( {\mathfrak {k}_{2k-2}^{\mathord {+}}\mathfrak {k}_{2k-1}}\right)$ and $\textbf{r}_{2k}\left( {\alpha }\right) =\left( {-1}\right) ^k\left( {\mathfrak {k}_{2k-1}\mathfrak {k}_{2k-2}^{\mathord {+}}}\right) \left( {\mathfrak {k}_{2k-3}\mathfrak {k}_{2k-4}^{\mathord {+}}}\right) \cdots \left( {\mathfrak {k}_{1}\mathfrak {k}_{0}^{\mathord {+}}}\right)$ for each $k\in \mathbb {N}$ fulfilling $2k\le \kappa +1$.

Lemma 10.10

Let $(s_j)_{j=0}^{\kappa }$ be a sequence of Hermitian complex ${q\times q}$ matrices and let $z\in \mathbb {C}$. For each $\ell \in \mathbb {Z}_{0,\kappa +1}$, then $\left[ {\textbf{p}_{\ell }\left( {\overline{z}}\right) } \right] ^*=\textbf{r}_{\ell }\left( {z}\right)$ and $\left[ {\textbf{q}_{\ell }\left( {\overline{z}}\right) } \right] ^*=\textbf{t}_{\ell }\left( {z}\right)$.

Proof

First observe that [8, Thm. 4.12(a)] yields $\mathfrak {k}_{j}^*=\mathfrak {k}_{j}$ for all $j\in \mathbb {Z}_{0,\kappa }$. Using Remark A.12, then $\left( {\mathfrak {k}_{j}^{\mathord {+}}}\right) ^*=\mathfrak {k}_{j}^{\mathord {+}}$ for all $j\in \mathbb {Z}_{0,\kappa }$ follows. From (10.2) we get $\left[ {\textbf{p}_{0}\left( {\overline{z}}\right) } \right] ^*=\textbf{r}_{0}\left( {z}\right)$ and $\left[ {\textbf{q}_{0}\left( {\overline{z}}\right) } \right] ^*=\textbf{t}_{0}\left( {z}\right)$. In view of (10.3), we have $\left[ {\textbf{p}_{1}\left( {\overline{z}}\right) } \right] ^*=\textbf{r}_{1}\left( {z}\right)$ and $\left[ {\textbf{q}_{1}\left( {\overline{z}}\right) } \right] ^*=\mathfrak {k}_{0}^*=\mathfrak {k}_{0}=\textbf{t}_{1}\left( {z}\right)$. Now assume $\kappa \ge 1$ and that there exists an integer $\ell \in \mathbb {Z}_{2,\kappa +1}$ such that $\left[ {\textbf{p}_{m}\left( {\overline{z}}\right) } \right] ^*=\textbf{r}_{m}\left( {z}\right)$ and $\left[ {\textbf{q}_{m}\left( {\overline{z}}\right) } \right] ^*=\textbf{t}_{m}\left( {z}\right)$ are valid for each $m\in \mathbb {Z}_{0,\ell -1}$. Taking additionally into account (10.4), Notation 10.6, and (10.5), we can conclude

$$\begin{aligned}\begin{aligned} \left[ {\textbf{p}_{\ell }\left( {\overline{z}}\right) } \right] ^*&=\left[ {\varepsilon _{\ell -1}\left( {\overline{z}}\right) \textbf{p}_{\ell -1}\left( {\overline{z}}\right) -\textbf{p}_{\ell -2}\left( {\overline{z}}\right) \mathfrak {k}_{\ell -2}^{\mathord {+}}\mathfrak {k}_{\ell -1}} \right] ^*\\&=\left[ {\varepsilon _{\ell -1}\left( {\overline{z}}\right) } \right] ^*\left[ {\textbf{p}_{\ell -1}\left( {\overline{z}}\right) } \right] ^*-\mathfrak {k}_{\ell -1}^*\left( {\mathfrak {k}_{\ell -2}^{\mathord {+}}}\right) ^*\left[ {\textbf{p}_{\ell -2}\left( {\overline{z}}\right) } \right] ^*\\&=\varepsilon _{\ell -1}\left( {z}\right) \textbf{r}_{\ell -1}\left( {z}\right) -\mathfrak {k}_{\ell -1}\mathfrak {k}_{\ell -2}^{\mathord {+}}\textbf{r}_{\ell -2}\left( {z}\right) =\textbf{r}_{\ell }\left( {z}\right) \end{aligned}\end{aligned}$$

and, analogously, $\left[ {\textbf{q}_{\ell }\left( {\overline{z}}\right) } \right] ^*=\textbf{t}_{\ell }\left( {z}\right)$. Thus, the assertion is proved inductively. $\square$

Remark 10.11

([13, Cor. 15.4]) Let $(s_j)_{j=0}^{\kappa }\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,\kappa ,\alpha }$ and let $\ell \in \mathbb {Z}_{0,\kappa +1}$. For all $z\in \mathbb {C}\backslash {[\alpha ,\infty )}$, then $\det \textbf{p}_{\ell }\left( {z}\right) \ne 0$.

Remark 10.12

Let $(s_j)_{j=0}^{\kappa }\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,\kappa ,\alpha }$ and let $\ell \in \mathbb {Z}_{0,\kappa +1}$. In view of Remark 10.11 and Lemma 10.10, then $\mathcal {Z}\left( {\det \textbf{r}_{\ell }}\right) =\mathcal {Z}\left( {\det \textbf{p}_{\ell }}\right) \subseteq {[\alpha ,\infty )}$.

Proposition 10.13

([13, Prop. 14.9]) Let $(s_j)_{j=0}^{\kappa }\in \mathcal {K}^\succcurlyeq _{q,\kappa ,\alpha }$. For each $\ell \in \mathbb {Z}_{0,\kappa +1}$, then $\textbf{q}_{\ell }=\textbf{p}_{\ell }^{\llbracket s\rrbracket }$.

Remark 10.14

In view of Notations 10.6 and 5.5, we have $\varepsilon _{m}=\varepsilon _{m+2}$ and $\varepsilon _{m}\varepsilon _{m+1}=\epsilon -\alpha$ for all $m\in \mathbb {N}_0$.

Lemma 10.15

Suppose $\kappa \ge 1$. Let $(s_j)_{j=0}^{\kappa }$ be a sequence of complex ${q\times q}$ matrices. Then $\textbf{p}_{2}=\left( {\epsilon -\alpha }\right) I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{1}$ and $\textbf{r}_{2}=\left( {\epsilon -\alpha }\right) I_{q}-\mathfrak {k}_{1}\mathfrak {k}_{0}^{\mathord {+}}$. If $\kappa \ge 2$, then $\textbf{p}_{3}=\textbf{p}_{1}\left[ {\left( {\epsilon -\alpha }\right) I_{q}-\left( {\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{1}+\mathfrak {k}_{1}^{\mathord {+}}\mathfrak {k}_{2}}\right) } \right]$ and $\textbf{r}_{3}=\left[ {\left( {\epsilon -\alpha }\right) I_{q}-\left( {\mathfrak {k}_{1}\mathfrak {k}_{0}^{\mathord {+}}+\mathfrak {k}_{2}\mathfrak {k}_{1}^{\mathord {+}}}\right) } \right] \textbf{r}_{1}$. If $\kappa \ge 3$, then, for all $\ell \in \mathbb {Z}_{4,\kappa +1}$, furthermore

$$\begin{aligned} \textbf{p}_{\ell } =\textbf{p}_{\ell -2}\left[ {\left( {\epsilon -\alpha }\right) I_{q}-\left( {\mathfrak {k}_{\ell -3}^{\mathord {+}}\mathfrak {k}_{\ell -2}+\mathfrak {k}_{\ell -2}^{\mathord {+}}\mathfrak {k}_{\ell -1}}\right) } \right] -\textbf{p}_{\ell -4}\mathfrak {k}_{\ell -4}^{\mathord {+}}\mathfrak {k}_{\ell -3}\mathfrak {k}_{\ell -3}^{\mathord {+}}\mathfrak {k}_{\ell -2} \nonumber \\ \end{aligned}$$

(10.6)

and

$$\begin{aligned} \textbf{r}_{\ell } =\left[ {\left( {\epsilon -\alpha }\right) I_{q}-\left( {\mathfrak {k}_{\ell -2}\mathfrak {k}_{\ell -3}^{\mathord {+}}+\mathfrak {k}_{\ell -1}\mathfrak {k}_{\ell -2}^{\mathord {+}}}\right) } \right] \textbf{r}_{\ell -2}-\mathfrak {k}_{\ell -2}\mathfrak {k}_{\ell -3}^{\mathord {+}}\mathfrak {k}_{\ell -3}\mathfrak {k}_{\ell -4}^{\mathord {+}}\textbf{r}_{\ell -4}.\nonumber \\ \end{aligned}$$

(10.7)

Proof

Using (10.4), Notation 10.6, (10.3), Notation 5.5, and (10.2), we obtain $\textbf{p}_{2}=\varepsilon _{1}\textbf{p}_{1}-\textbf{p}_{0}\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{1}=\left( {\epsilon -\alpha }\right) I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{1}$ and, with (10.5) instead of (10.4), analogously $\textbf{r}_{2}=\left( {\epsilon -\alpha }\right) I_{q}-\mathfrak {k}_{1}\mathfrak {k}_{0}^{\mathord {+}}$. Now suppose $\kappa \ge 2$. Then we obtain similarly

$$\begin{aligned}\begin{aligned} \textbf{p}_{3}&=\varepsilon _{2}\textbf{p}_{2}-\textbf{p}_{1}\mathfrak {k}_{1}^{\mathord {+}}\mathfrak {k}_{2}\\&=\left( {\epsilon -\alpha }\right) I_{q}\cdot \left[ {\left( {\epsilon -\alpha }\right) I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{1}-\mathfrak {k}_{1}^{\mathord {+}}\mathfrak {k}_{2}} \right] =\textbf{p}_{1}\left[ {\left( {\epsilon -\alpha }\right) I_{q}-\left( {\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{1}+\mathfrak {k}_{1}^{\mathord {+}}\mathfrak {k}_{2}}\right) } \right] \end{aligned}\end{aligned}$$

and, analogously, $\textbf{r}_{3}=\left[ {\left( {\epsilon -\alpha }\right) I_{q}-\left( {\mathfrak {k}_{1}\mathfrak {k}_{0}^{\mathord {+}}+\mathfrak {k}_{2}\mathfrak {k}_{1}^{\mathord {+}}}\right) } \right] \textbf{r}_{1}$. Now suppose $\kappa \ge 3$. Let $\ell \in \mathbb {Z}_{4,\kappa +1}$. From (10.4) we can infer then $\varepsilon _{\ell -3}\textbf{p}_{\ell -3}=\textbf{p}_{\ell -2}+\textbf{p}_{\ell -4}\mathfrak {k}_{\ell -4}^{\mathord {+}}\mathfrak {k}_{\ell -3}$. Using additionally (10.4) twice and Remark 10.14, we can conclude

$$\begin{aligned}\begin{aligned} \textbf{p}_{\ell }&=\varepsilon _{\ell -1}\textbf{p}_{\ell -1}-\textbf{p}_{\ell -2}\mathfrak {k}_{\ell -2}^{\mathord {+}}\mathfrak {k}_{\ell -1}\\&=\varepsilon _{\ell -1}\left( {\varepsilon _{\ell -2}\textbf{p}_{\ell -2}-\textbf{p}_{\ell -3}\mathfrak {k}_{\ell -3}^{\mathord {+}}\mathfrak {k}_{\ell -2}}\right) -\textbf{p}_{\ell -2}\mathfrak {k}_{\ell -2}^{\mathord {+}}\mathfrak {k}_{\ell -1}\\&=\varepsilon _{\ell -1}\varepsilon _{\ell -2}\textbf{p}_{\ell -2}-\varepsilon _{\ell -1}\textbf{p}_{\ell -3}\mathfrak {k}_{\ell -3}^{\mathord {+}}\mathfrak {k}_{\ell -2}-\textbf{p}_{\ell -2}\mathfrak {k}_{\ell -2}^{\mathord {+}}\mathfrak {k}_{\ell -1}\\&=\left( {\epsilon -\alpha }\right) \textbf{p}_{\ell -2}-\varepsilon _{\ell -3}\textbf{p}_{\ell -3}\mathfrak {k}_{\ell -3}^{\mathord {+}}\mathfrak {k}_{\ell -2}-\textbf{p}_{\ell -2}\mathfrak {k}_{\ell -2}^{\mathord {+}}\mathfrak {k}_{\ell -1}\\&=\left( {\epsilon -\alpha }\right) \textbf{p}_{\ell -2}-\left( {\textbf{p}_{\ell -2}+\textbf{p}_{\ell -4}\mathfrak {k}_{\ell -4}^{\mathord {+}}\mathfrak {k}_{\ell -3}}\right) \mathfrak {k}_{\ell -3}^{\mathord {+}}\mathfrak {k}_{\ell -2}-\textbf{p}_{\ell -2}\mathfrak {k}_{\ell -2}^{\mathord {+}}\mathfrak {k}_{\ell -1}\\&=\textbf{p}_{\ell -2}\left[ {\left( {\epsilon -\alpha }\right) I_{q}-\left( {\mathfrak {k}_{\ell -3}^{\mathord {+}}\mathfrak {k}_{\ell -2}+\mathfrak {k}_{\ell -2}^{\mathord {+}}\mathfrak {k}_{\ell -1}}\right) } \right] -\textbf{p}_{\ell -4}\mathfrak {k}_{\ell -4}^{\mathord {+}}\mathfrak {k}_{\ell -3}\mathfrak {k}_{\ell -3}^{\mathord {+}}\mathfrak {k}_{\ell -2}. \end{aligned}\end{aligned}$$

Thus, (10.6) is proved. Analogously, from (10.5) we can obtain (10.7). We omit the details. $\square$

Lemma 10.16

Suppose $\kappa \ge 1$. Let $(s_j)_{j=0}^{\kappa }$ be a sequence of complex ${q\times q}$ matrices. For all $z\in \mathbb {C}$, then $\textbf{q}_{2}\left( {z}\right) =\mathfrak {k}_{0}$ and $\textbf{t}_{2}\left( {z}\right) =\mathfrak {k}_{0}$. If $\kappa \ge 2$, then $\textbf{q}_{3}=\textbf{q}_{1}\left[ {\left( {\epsilon -\alpha }\right) I_{q}-\mathfrak {k}_{1}^{\mathord {+}}\mathfrak {k}_{2}} \right]$ and $\textbf{t}_{3}=\left[ {\left( {\epsilon -\alpha }\right) I_{q}-\mathfrak {k}_{2}\mathfrak {k}_{1}^{\mathord {+}}} \right] \textbf{t}_{1}$. If $\kappa \ge 3$, then, for all $\ell \in \mathbb {Z}_{4,\kappa +1}$, furthermore

$$\begin{aligned} \textbf{q}_{\ell }&=\textbf{q}_{\ell -2}\left[ {\left( {\epsilon -\alpha }\right) I_{q}-\left( {\mathfrak {k}_{\ell -3}^{\mathord {+}}\mathfrak {k}_{\ell -2}+\mathfrak {k}_{\ell -2}^{\mathord {+}}\mathfrak {k}_{\ell -1}}\right) } \right] -\textbf{q}_{\ell -4}\mathfrak {k}_{\ell -4}^{\mathord {+}}\mathfrak {k}_{\ell -3}\mathfrak {k}_{\ell -3}^{\mathord {+}}\mathfrak {k}_{\ell -2} \end{aligned}$$

and

$$\begin{aligned} \textbf{t}_{\ell }&=\left[ {\left( {\epsilon -\alpha }\right) I_{q}-\left( {\mathfrak {k}_{\ell -2}\mathfrak {k}_{\ell -3}^{\mathord {+}}+\mathfrak {k}_{\ell -1}\mathfrak {k}_{\ell -2}^{\mathord {+}}}\right) } \right] \textbf{t}_{\ell -2}-\mathfrak {k}_{\ell -2}\mathfrak {k}_{\ell -3}^{\mathord {+}}\mathfrak {k}_{\ell -3}\mathfrak {k}_{\ell -4}^{\mathord {+}}\textbf{t}_{\ell -4}. \end{aligned}$$

Lemma 10.16 can be proved analogous to Lemma 10.15. We omit the details. With regard to the following theorems we refer to Notation 8.4 and Definition 8.1.

Theorem 10.17

([13, Thm. 15.6]) Let $n\in \mathbb {N}_0$ and let $(s_j)_{j=0}^{2n}\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,2n,\alpha }$. Let $\tilde{\textbf{p}}_{2n}^\flat ,\tilde{\textbf{q}}_{2n}^\flat :\mathbb {C}\backslash {[\alpha ,\infty )}\rightarrow \mathbb {C}^{{q\times q}}$ be defined by

$$\begin{aligned} \tilde{\textbf{p}}_{2n}^\flat \left( {z}\right)&:=\left( {z-\alpha }\right) \textbf{p}_{2n}\left( {z}\right){} & {} \text {and}&\tilde{\textbf{q}}_{2n}^\flat \left( {z}\right)&:=\left( {z-\alpha }\right) \textbf{q}_{2n}\left( {z}\right) . \end{aligned}$$

Denote by $\tilde{\textbf{p}}_{2n+1}$ and $\tilde{\textbf{q}}_{2n+1}$ the restriction of $\textbf{p}_{2n+1}$ and $\textbf{q}_{2n+1}$ onto $\mathbb {C}\backslash {[\alpha ,\infty )}$, respectively.

• (a) Let $\Gamma \in \langle {\mathcal {P}_{{\mathord {\bullet }},\alpha }\left[ {L_{n}} \right] } \rangle$ and let $\left( {G_1};{G_2}\right) \in \Gamma$. Then $\det \left( {\tilde{\textbf{p}}_{2n}^\flat L_{n}^{\mathord {+}}G_1+\tilde{\textbf{p}}_{2n+1}G_2}\right)$ does not vanish identically in $\mathbb {C}\backslash {[\alpha ,\infty )}$ and F given by

  $$\begin{aligned} F =-\left( {\tilde{\textbf{q}}_{2n}^\flat L_{n}^{\mathord {+}}G_1+\tilde{\textbf{q}}_{2n+1}G_2}\right) \left( {\tilde{\textbf{p}}_{2n}^\flat L_{n}^{\mathord {+}}G_1+\tilde{\textbf{p}}_{2n+1}G_2}\right) ^{-1}\end{aligned}$$

  (10.8)

  belongs to ${\mathcal {S}_{0,q}\left[ {\mathbb {C}\backslash {[\alpha ,\infty )};(s_j)_{j=0}^{2n},\preccurlyeq } \right] }$.

• (b) For each $F\in {\mathcal {S}_{0,q}\left[ {\mathbb {C}\backslash {[\alpha ,\infty )};(s_j)_{j=0}^{2n},\preccurlyeq } \right] }$, there exists a unique equivalence class $\Gamma \in \langle {\mathcal {P}_{{\mathord {\bullet }},\alpha }\left[ {L_{n}} \right] } \rangle$ such that (10.8) is fulfilled for each $\left( {G_1};{G_2}\right) \in \Gamma$.

Theorem 10.18

([13, Thm. 15.7]) Let $n\in \mathbb {N}_0$ and let $(s_j)_{j=0}^{2n+1}\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,2n+1,\alpha }$. Denote by $\tilde{\textbf{p}}_{2n+1}$, $\tilde{\textbf{q}}_{2n+1}$, $\tilde{\textbf{p}}_{2n+2}$, and $\tilde{\textbf{q}}_{2n+2}$ the restriction of $\textbf{p}_{2n+1}$, $\textbf{q}_{2n+1}$, $\textbf{p}_{2n+2}$, and $\textbf{q}_{2n+2}$ onto $\mathbb {C}\backslash {[\alpha ,\infty )}$, respectively.

• (a) Let $\Gamma \in \langle {\mathcal {P}_{{\mathord {\bullet }},\alpha }\left[ {L_{{\alpha ,n}}} \right] } \rangle$ and let $\left( {G_1};{G_2}\right) \in \Gamma$. Then $\det \left( {\tilde{\textbf{p}}_{2n+1}L_{{\alpha ,n}}^{\mathord {+}}G_1+\tilde{\textbf{p}}_{2n+2}G_2}\right)$ does not vanish identically in $\mathbb {C}\backslash {[\alpha ,\infty )}$ and F given by

  $$\begin{aligned} F =-\left( {\tilde{\textbf{q}}_{2n+1}L_{{\alpha ,n}}^{\mathord {+}}G_1+\tilde{\textbf{q}}_{2n+2}G_2}\right) \left( {\tilde{\textbf{p}}_{2n+1}L_{{\alpha ,n}}^{\mathord {+}}G_1+\tilde{\textbf{p}}_{2n+2}G_2}\right) ^{-1}\end{aligned}$$

  (10.9)

  belongs to ${\mathcal {S}_{0,q}\left[ {\mathbb {C}\backslash {[\alpha ,\infty )};(s_j)_{j=0}^{2n+1},\preccurlyeq } \right] }$.

• (b) For each $F\in {\mathcal {S}_{0,q}\left[ {\mathbb {C}\backslash {[\alpha ,\infty )};(s_j)_{j=0}^{2n+1},\preccurlyeq } \right] }$, there exists a unique equivalence class $\Gamma \in \langle {\mathcal {P}_{{\mathord {\bullet }},\alpha }\left[ {L_{{\alpha ,n}}} \right] } \rangle$ such that (10.9) is fulfilled for each $\left( {G_1};{G_2}\right) \in \Gamma$.

11 The First ${[\alpha ,\infty )}$-quadruple of Matrix Polynomials

We now study in detail a subsystem of the quadruple of matrix polynomials introduced in the previous section. This section is also primarily technical in nature. We point out that in Lemma 11.11 and Proposition 11.13 connections are made with pairs of meromorphic functions introduced above.

Remark 11.1

(cf. [13, Notation  14.5]) Let $(s_j)_{j=0}^{\kappa }$ be a sequence of complex ${q\times q}$ matrices. In view of Remark 10.8, for each $k\in \mathbb {N}_0$ fulfilling $2k\le \kappa +1$, the functions

$$\begin{aligned} \textbf{b}_{k}:=\textbf{p}_{2k}\quad \text {and}\quad \textbf{d}_{k}:=\textbf{r}_{2k} \end{aligned}$$

(11.1)

are ${q\times q}$ matrix polynomials with degree k and leading coefficient matrix $I_{q}$. Taking into account Notation 10.3, it is readily checked that, for each $k\in \mathbb {N}_0$ fulfilling $2k\le \kappa +1$, then

$$\begin{aligned} \textbf{a}_{k}:=\textbf{b}_{k}^{\llbracket s\rrbracket }\quad \text {and}\quad \textbf{c}_{k}:=\textbf{d}_{k}^{\langle s\rangle } \end{aligned}$$

(11.2)

are ${q\times q}$ matrix polynomials of degree not greater than $k-1$. The quadruple $\left[ {(\textbf{a}_{k})_{k=0}^{\dot{\kappa }},(\textbf{b}_{k})_{k=0}^{\dot{\kappa }},(\textbf{c}_{k})_{k=0}^{\dot{\kappa }},(\textbf{d}_{k})_{k=0}^{\dot{\kappa }}} \right]$ will be called the first ${[\alpha ,\infty )}$-quadruple of matrix polynomials (short: first ${[\alpha ,\infty )}$-QMP) associated with $(s_j)_{j=0}^{\kappa }$.

Remark 11.2

Let $(s_j)_{j=0}^{\kappa }$ be a sequence of Hermitian complex ${q\times q}$ matrices and let $k\in \mathbb {N}_0$ be such that $2k\le \kappa +1$. In view of (11.1) and Lemma 10.10, then $\left[ {\textbf{b}_{k}\left( {\overline{z}}\right) } \right] ^*=\textbf{d}_{k}\left( {z}\right)$ for all $z\in \mathbb {C}$. According to (2.2), hence $\textbf{b}_{k}^{\mathord {\vee }}=\textbf{d}_{k}$. In view of (11.2) and Lemma 10.4, then $\textbf{a}_{k}^{\mathord {\vee }}=\left( {\textbf{b}_{k}^{\llbracket s\rrbracket }}\right) ^{\mathord {\vee }}=\left( {\textbf{b}_{k}^{\mathord {\vee }}}\right) ^{\langle s\rangle }=\textbf{d}_{k}^{\langle s\rangle }=\textbf{c}_{k}$. According to (2.2), thus $\left[ {\textbf{a}_{k}\left( {\overline{z}}\right) } \right] ^*=\textbf{c}_{k}\left( {z}\right)$ for all $z\in \mathbb {C}$.

Lemma 11.3

Let $(s_j)_{j=0}^{\kappa }\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,\kappa ,\alpha }$ and let $k\in \mathbb {N}_0$ be such that $2k\le \kappa +1$. Then $\textbf{a}_{k}=\textbf{q}_{2k}$ and $\textbf{c}_{k}=\textbf{t}_{2k}$.

Proof

Using (11.2) and Proposition 10.13, we get $\textbf{a}_{k}=\textbf{b}_{k}^{\llbracket s\rrbracket }=\textbf{p}_{2k}^{\llbracket s\rrbracket }=\textbf{q}_{2k}$. Since Remark 9.3 shows that $(s_j)_{j=0}^{\kappa }$ is a sequence of Hermitian matrices, we can additionally use Remark 11.2 and Lemma 10.10 to get $\textbf{c}_{k}\left( {z}\right) =\left[ {\textbf{a}_{k}\left( {\overline{z}}\right) } \right] ^*=\left[ {\textbf{q}_{2k}\left( {\overline{z}}\right) } \right] ^*=\textbf{t}_{2k}\left( {z}\right)$ for all $z\in \mathbb {C}$. $\square$

Remark 11.4

Let $(s_j)_{j=0}^{\kappa }\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,\kappa ,\alpha }$ and let $k\in \mathbb {N}_0$ be such that $2k\le \kappa +1$. In view of (11.1) and Remark 10.12, then $\mathcal {Z}\left( {\det \textbf{d}_{k}}\right) =\mathcal {Z}\left( {\det \textbf{b}_{k}}\right) \subseteq {[\alpha ,\infty )}$.

Lemma 11.5

Let $(s_j)_{j=0}^{\kappa }\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,\kappa ,\alpha }$. For all $z\in \mathbb {C}$, then $\textbf{b}_{0}\left( {z}\right) =I_{q}$ and $\textbf{d}_{0}\left( {z}\right) =I_{q}$. If $\kappa \ge 1$, then $\textbf{b}_{1}=\left( {\epsilon -\alpha }\right) I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{1}$ and $\textbf{d}_{1}=\left( {\epsilon -\alpha }\right) I_{q}-\mathfrak {k}_{1}\mathfrak {k}_{0}^{\mathord {+}}$. If $\kappa \ge 3$, for all $k\in \mathbb {Z}_{2,\infty }$ fulfilling $2k\le \kappa +1$, furthermore

$$\begin{aligned} \textbf{b}_{k} =\textbf{b}_{k-1}\left[ {\left( {\epsilon -\alpha }\right) I_{q}-\left( {\mathfrak {k}_{2k-3}^{\mathord {+}}\mathfrak {k}_{2k-2}+\mathfrak {k}_{2k-2}^{\mathord {+}}\mathfrak {k}_{2k-1}}\right) } \right] -\textbf{b}_{k-2}\mathfrak {k}_{2k-4}^{\mathord {+}}\mathfrak {k}_{2k-2}\qquad \end{aligned}$$

(11.3)

and

$$\begin{aligned} \textbf{d}_{k} =\left[ {\left( {\epsilon -\alpha }\right) I_{q}-\left( {\mathfrak {k}_{2k-2}\mathfrak {k}_{2k-3}^{\mathord {+}}+\mathfrak {k}_{2k-1}\mathfrak {k}_{2k-2}^{\mathord {+}}}\right) } \right] \textbf{d}_{k-1}-\mathfrak {k}_{2k-2}\mathfrak {k}_{2k-4}^{\mathord {+}}\textbf{d}_{k-2}.\qquad \end{aligned}$$

(11.4)

Proof

In view of (11.1) and (10.2), we have $\textbf{b}_{0}\left( {z}\right) =I_{q}$ and $\textbf{d}_{0}\left( {z}\right) =I_{q}$ for all $z\in \mathbb {C}$. If $\kappa \ge 1$, we can use (11.1) and Lemma 10.15 to obtain $\textbf{b}_{1}=\left( {\epsilon -\alpha }\right) I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{1}$ and $\textbf{d}_{1}=\left( {\epsilon -\alpha }\right) I_{q}-\mathfrak {k}_{1}\mathfrak {k}_{0}^{\mathord {+}}$. Now suppose $\kappa \ge 3$. Let $k\in \mathbb {Z}_{2,\infty }$ be such that $2k\le \kappa +1$. Remark 9.3 yields then $\mathfrak {k}_{2k-3}\mathfrak {k}_{2k-3}^{\mathord {+}}\mathfrak {k}_{2k-2}=\mathfrak {k}_{2k-2}$ and $\mathfrak {k}_{2k-2}\mathfrak {k}_{2k-3}^{\mathord {+}}\mathfrak {k}_{2k-3}=\mathfrak {k}_{2k-2}$. Taking additionally into account (11.1) and Lemma 10.15, we can conclude then (11.3) and (11.4). $\square$

Lemma 11.6

Let $(s_j)_{j=0}^{\kappa }\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,\kappa ,\alpha }$. For all $z\in \mathbb {C}$, then $\textbf{a}_{0}\left( {z}\right) =O_{{q\times q}}$ and $\textbf{c}_{0}\left( {z}\right) =O_{{q\times q}}$. If $\kappa \ge 1$, then $\textbf{a}_{1}\left( {z}\right) =\mathfrak {k}_{0}$ and $\textbf{c}_{1}\left( {z}\right) =\mathfrak {k}_{0}$ for all $z\in \mathbb {C}$. If $\kappa \ge 3$, for all $k\in \mathbb {Z}_{2,\infty }$ fulfilling $2k\le \kappa +1$, furthermore

$$\begin{aligned} \textbf{a}_{k}=\textbf{a}_{k-1}\left[ {\left( {\epsilon -\alpha }\right) I_{q}-\left( {\mathfrak {k}_{2k-3}^{\mathord {+}}\mathfrak {k}_{2k-2}+\mathfrak {k}_{2k-2}^{\mathord {+}}\mathfrak {k}_{2k-1}}\right) } \right] -\textbf{a}_{k-2}\mathfrak {k}_{2k-4}^{\mathord {+}}\mathfrak {k}_{2k-2} \end{aligned}$$

and

$$\begin{aligned} \textbf{c}_{k}=\left[ {\left( {\epsilon -\alpha }\right) I_{q}-\left( {\mathfrak {k}_{2k-2}\mathfrak {k}_{2k-3}^{\mathord {+}}+\mathfrak {k}_{2k-1}\mathfrak {k}_{2k-2}^{\mathord {+}}}\right) } \right] \textbf{c}_{k-1}-\mathfrak {k}_{2k-2}\mathfrak {k}_{2k-4}^{\mathord {+}}\textbf{c}_{k-2}. \end{aligned}$$

Using Lemmata 11.3 and 10.16, (10.2), and Remark 9.3, one can prove Lemma 11.6 in the same way as Lemma 11.5. We omit the details.

Remark 11.7

Suppose $\kappa \ge 3$. Let $(s_j)_{j=0}^{\kappa }\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,\kappa ,\alpha }$ and let $k\in \mathbb {N}$ be such that $2k+1\le \kappa$. In view of Lemmata 11.6 and 11.5, then

and

Lemma 11.8

Suppose $\kappa \ge 1$. Let $(s_j)_{j=0}^{\kappa }\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,\kappa ,\alpha }$ and let $z\in \mathbb {C}\backslash {[\alpha ,\infty )}$. For each $k\in \mathbb {N}_0$ fulfilling $2k+1\le \kappa$, then

$$\begin{aligned} \det \textbf{b}_{k}\left( {z}\right) \ne 0,\, \det \textbf{d}_{k}\left( {z}\right) \ne 0,\, \det \textbf{b}_{k+1}\left( {z}\right) \ne 0, \text { and } \det \textbf{d}_{k+1}\left( {z}\right) \ne 0 \end{aligned}$$

(11.5)

as well as

$$\begin{aligned} \mathfrak {h}_{2k}\left[ {\textbf{b}_{k}\left( {z}\right) } \right] ^{-1}\textbf{b}_{k+1}\left( {z}\right)&=\textbf{d}_{k+1}\left( {z}\right) \left[ {\textbf{d}_{k}\left( {z}\right) } \right] ^{-1}\mathfrak {h}_{2k} \end{aligned}$$

(11.6)

and

$$\begin{aligned} \mathfrak {h}_{2k}\left[ {\textbf{b}_{k+1}\left( {z}\right) } \right] ^{-1}\textbf{b}_{k}\left( {z}\right)&=\textbf{d}_{k}\left( {z}\right) \left[ {\textbf{d}_{k+1}\left( {z}\right) } \right] ^{-1}\mathfrak {h}_{2k}. \end{aligned}$$

(11.7)

Proof

From Remark 11.4 we know that (11.5) is valid for each $k\in \mathbb {N}_0$ fulfilling $2k+1\le \kappa$. We now proceed by mathematical induction. By virtue of Lemma 11.5, Notation 5.5, and Lemma 9.5, we obtain $\mathfrak {h}_{0}\left[ {\textbf{b}_{0}\left( {z}\right) } \right] ^{-1}\textbf{b}_{1}\left( {z}\right) =z\mathfrak {h}_{0}-\mathfrak {h}_{0}\left( {\alpha I_{q}+\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{1}}\right) =z\mathfrak {h}_{0}-\mathfrak {h}_{1}$ and, analogously, $\textbf{d}_{1}\left( {z}\right) \left[ {\textbf{d}_{0}\left( {z}\right) } \right] ^{-1}\mathfrak {h}_{0}=z\mathfrak {h}_{0}-\mathfrak {h}_{1}$. Therefore, (11.6) follows for $k=0$, which in turn implies (11.7) for $k=0$. Now suppose $\kappa \ge 3$ and that there exists an integer $\ell \in \mathbb {N}$ fulfilling $2\ell +1\le \kappa$ such that (11.6) and (11.7) are valid for $k=\ell -1$. Using Lemma 11.5, Notation 5.5, and Lemma 9.5, we obtain

$$\begin{aligned}&\mathfrak {h}_{2\ell }\left[ {\textbf{b}_{\ell }\left( {z}\right) } \right] ^{-1}\textbf{b}_{\ell +1}\left( {z}\right) \nonumber \\&\quad =\mathfrak {h}_{2\ell }\left[ {\textbf{b}_{\ell }\left( {z}\right) } \right] ^{-1}\left( {\textbf{b}_{\ell }\left( {z}\right) \left[ {\left( {z-\alpha }\right) I_{q}-\left( {\mathfrak {k}_{2\ell -1}^{\mathord {+}}\mathfrak {k}_{2\ell }+\mathfrak {k}_{2\ell }^{\mathord {+}}\mathfrak {k}_{2\ell +1}}\right) } \right] -\textbf{b}_{\ell -1}\left( {z}\right) \mathfrak {k}_{2\ell -2}^{\mathord {+}}\mathfrak {k}_{2\ell }}\right) \nonumber \\&\quad =z\mathfrak {h}_{2\ell }-\mathfrak {h}_{2\ell }\left( {\alpha I_{q}+\mathfrak {k}_{2\ell -1}^{\mathord {+}}\mathfrak {k}_{2\ell }+\mathfrak {k}_{2\ell }^{\mathord {+}}\mathfrak {k}_{2\ell +1}}\right) -\mathfrak {h}_{2\ell }\left[ {\textbf{b}_{\ell }\left( {z}\right) } \right] ^{-1}\textbf{b}_{\ell -1}\left( {z}\right) \mathfrak {k}_{2\ell -2}^{\mathord {+}}\mathfrak {k}_{2\ell }\nonumber \\&\quad =z\mathfrak {h}_{2\ell }-\mathfrak {h}_{2\ell +1}-\mathfrak {h}_{2\ell }\left[ {\textbf{b}_{\ell }\left( {z}\right) } \right] ^{-1}\textbf{b}_{\ell -1}\left( {z}\right) \mathfrak {k}_{2\ell -2}^{\mathord {+}}\mathfrak {k}_{2\ell } \end{aligned}$$

(11.8)

and, analogously, $\textbf{d}_{\ell +1}\left( {z}\right) \left[ {\textbf{d}_{\ell }\left( {z}\right) } \right] ^{-1}\mathfrak {h}_{2\ell }=z\mathfrak {h}_{2\ell }-\mathfrak {h}_{2\ell +1}-\mathfrak {k}_{2\ell }\mathfrak {k}_{2\ell -2}^{\mathord {+}}\textbf{d}_{\ell -1}\left( {z}\right) \left[ {\textbf{d}_{\ell }\left( {z}\right) } \right] ^{-1}\mathfrak {h}_{2\ell }$. In view of Remark 7.6, we have $(s_j)_{j=0}^{\kappa }\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,\kappa }$. Thus, Remark 5.3 yields $\mathfrak {h}_{2\ell -2}\mathfrak {h}_{2\ell -2}^{\mathord {+}}\mathfrak {h}_{2\ell }=\mathfrak {h}_{2\ell }$ and $\mathfrak {h}_{2\ell }\mathfrak {h}_{2\ell -2}^{\mathord {+}}\mathfrak {h}_{2\ell -2}=\mathfrak {h}_{2\ell }$. Furthermore, Remark 9.2 provides $\mathfrak {k}_{2\ell -2}=\mathfrak {h}_{2\ell -2}$ and $\mathfrak {k}_{2\ell }=\mathfrak {h}_{2\ell }$. Taking additionally into account (11.7) for $k=\ell -1$, we can conclude

$$\begin{aligned}{} & {} \mathfrak {h}_{2\ell }\left[ {\textbf{b}_{\ell }\left( {z}\right) } \right] ^{-1}\textbf{b}_{\ell -1}\left( {z}\right) \mathfrak {k}_{2\ell -2}^{\mathord {+}}\mathfrak {k}_{2\ell } =\mathfrak {h}_{2\ell }\mathfrak {h}_{2\ell -2}^{\mathord {+}}\mathfrak {h}_{2\ell -2}\left[ {\textbf{b}_{\ell }\left( {z}\right) } \right] ^{-1}\textbf{b}_{\ell -1}\left( {z}\right) \mathfrak {h}_{2\ell -2}^{\mathord {+}}\mathfrak {h}_{2\ell }\\{} & {} \quad =\mathfrak {h}_{2\ell }\mathfrak {h}_{2\ell -2}^{\mathord {+}}\textbf{d}_{\ell -1}\left( {z}\right) \left[ {\textbf{d}_{\ell }\left( {z}\right) } \right] ^{-1}\mathfrak {h}_{2\ell -2}\mathfrak {h}_{2\ell -2}^{\mathord {+}}\mathfrak {h}_{2\ell } =\mathfrak {k}_{2\ell }\mathfrak {k}_{2\ell -2}^{\mathord {+}}\textbf{d}_{\ell -1}\left( {z}\right) \left[ {\textbf{d}_{\ell }\left( {z}\right) } \right] ^{-1}\mathfrak {h}_{2\ell }. \end{aligned}$$

Therefore, (11.6) follows for $k=\ell$, which in turn implies (11.7) for $k=\ell$. Thus, (11.6) and (11.7) are proved for each $k\in \mathbb {N}_0$ fulfilling $2k+1\le \kappa$ by mathematical induction. $\square$

Lemma 11.9

Suppose $\kappa \ge 1$. Let $(s_j)_{j=0}^{\kappa }\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,\kappa ,\alpha }$, let $k\in \mathbb {N}_0$ be such that $2k+1\le \kappa$, and let $z\in \mathbb {C}\backslash {[\alpha ,\infty )}$. Then $\det \textbf{b}_{k}\left( {z}\right) \ne 0$ and $\det \textbf{b}_{k+1}\left( {z}\right) \ne 0$ as well as $\left( {\mathfrak {h}_{2k}\left[ {\textbf{b}_{k}\left( {z}\right) } \right] ^{-1}\textbf{b}_{k+1}\left( {z}\right) }\right) ^{\mathord {+}}=\mathfrak {h}_{2k}^{\mathord {+}}\mathfrak {h}_{2k}\left[ {\textbf{b}_{k+1}\left( {z}\right) } \right] ^{-1}\textbf{b}_{k}\left( {z}\right) \mathfrak {h}_{2k}^{\mathord {+}}$.

Proof

In view of Lemma 11.8, all the matrices $B_{k}:=\textbf{b}_{k}\left( {z}\right)$, $D_{k}:=\textbf{d}_{k}\left( {z}\right)$, $B_{ k +1}:=\textbf{b}_{ k +1}\left( {z}\right)$, and $D_{ k +1}:=\textbf{d}_{ k +1}\left( {z}\right)$ are invertible. Hence, the matrices $L:=D_{k}D_{ k +1}^{-1}$ and $R:=B_{ k +1}^{-1}B_{k}$ are invertible. Setting $M:=\mathfrak {h}_{2 k }$, $N:=LMR^{-1}$, and $X:=MR^{-1}$, we thus can apply Lemma A.15 to obtain $X^{\mathord {+}}=N^{\mathord {+}}NRM^{\mathord {+}}$. Since Lemma 11.8 provides $MR=LM$, we have $N=MRR^{-1}=M$ and, consequently, $X^{\mathord {+}}=M^{\mathord {+}}MB_{ k +1}^{-1}B_{k}M^{\mathord {+}}$. In view of $X=MR^{-1}=MB_{k}^{-1}B_{ k +1}$, the proof is complete. $\square$

Lemma 11.10

Suppose $\kappa \ge 1$. Let $(s_j)_{j=0}^{\kappa }\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,\kappa ,\alpha }$ and let $z\in \mathbb {C}\backslash {[\alpha ,\infty )}$. For each $k\in \mathbb {N}_0$ fulfilling $2k+1\le \kappa$, then $\det \textbf{b}_{k}\left( {z}\right) \ne 0$ and

$$\begin{aligned} \chi _{2k+1}\left( {z}\right) =\mathfrak {h}_{2k}\left[ {\textbf{b}_{k}\left( {z}\right) } \right] ^{-1}\textbf{b}_{k+1}\left( {z}\right) . \end{aligned}$$

(11.9)

Proof

First observe that Remark 7.6 yields $(s_j)_{j=0}^{\kappa }\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,\kappa }$. Remark 11.4 shows $\det \textbf{b}_{n}\left( {z}\right) \ne 0$ for each $n\in \mathbb {N}_0$ fulfilling $2n\le \kappa +1$. We now proceed by mathematical induction. As in the proof of Lemma 11.8 we can infer $\mathfrak {h}_{0}\left[ {\textbf{b}_{0}\left( {z}\right) } \right] ^{-1}\textbf{b}_{1}\left( {z}\right) =z\mathfrak {h}_{0}-\mathfrak {h}_{1}$, which in view of (6.1) implies (11.9) for $k=0$. Now suppose $\kappa \ge 3$ and that there exists an integer $\ell \in \mathbb {N}$ fulfilling $2\ell +1\le \kappa$ such that (11.9) is valid for $k=\ell -1$. As in the proof of Lemma 11.8 we can infer that (11.8) holds true, whereas from [6, Lem. 7.13(b)] we obtain

$$\begin{aligned} \chi _{2\ell +1}\left( {z}\right) =z\mathfrak {h}_{2\ell }-\mathfrak {h}_{2\ell +1}-\mathfrak {h}_{2\ell }\left[ {\chi _{2\ell -1}\left( {z}\right) } \right] ^{\mathord {+}}\mathfrak {h}_{2\ell }. \end{aligned}$$

(11.10)

Remark 5.3 yields $\mathfrak {h}_{2\ell }\mathfrak {h}_{2\ell -2}^{\mathord {+}}\mathfrak {h}_{2\ell -2}=\mathfrak {h}_{2\ell }$. Remark 9.2 shows $\mathfrak {k}_{2\ell -2}=\mathfrak {h}_{2\ell -2}$ and $\mathfrak {k}_{2\ell }=\mathfrak {h}_{2\ell }$. Because of Lemma 11.9, we have $\left( {\mathfrak {h}_{2\ell -2}\left[ {\textbf{b}_{\ell -1}\left( {z}\right) } \right] ^{-1}\textbf{b}_{\ell }\left( {z}\right) }\right) ^{\mathord {+}}=\mathfrak {h}_{2\ell -2}^{\mathord {+}}\mathfrak {h}_{2\ell -2}\left[ {\textbf{b}_{\ell }\left( {z}\right) } \right] ^{-1}\textbf{b}_{\ell -1}\left( {z}\right) \mathfrak {h}_{2\ell -2}^{\mathord {+}}$. Thus, taking additionally into account (11.9) for $k=\ell -1$, we conclude

$$\begin{aligned}&\mathfrak {h}_{2\ell }\left[ {\textbf{b}_{\ell }\left( {z}\right) } \right] ^{-1}\textbf{b}_{\ell -1}\left( {z}\right) \mathfrak {k}_{2\ell -2}^{\mathord {+}}\mathfrak {k}_{2\ell } =\mathfrak {h}_{2\ell }\mathfrak {h}_{2\ell -2}^{\mathord {+}}\mathfrak {h}_{2\ell -2}\left[ {\textbf{b}_{\ell }\left( {z}\right) } \right] ^{-1}\textbf{b}_{\ell -1}\left( {z}\right) \mathfrak {h}_{2\ell -2}^{\mathord {+}}\mathfrak {h}_{2\ell } \nonumber \\&\quad =\mathfrak {h}_{2\ell }\left( {\mathfrak {h}_{2\ell -2}\left[ {\textbf{b}_{\ell -1}\left( {z}\right) } \right] ^{-1}\textbf{b}_{\ell }\left( {z}\right) }\right) ^{\mathord {+}}\mathfrak {h}_{2\ell } =\mathfrak {h}_{2\ell }\left[ {\chi _{2\ell -1}\left( {z}\right) } \right] ^{\mathord {+}}\mathfrak {h}_{2\ell }. \end{aligned}$$

(11.11)

Therefore, comparing (11.8), (11.10), and (11.11), we get (11.9) for $k=\ell$. Thus, (11.9) is proved for each $k\in \mathbb {N}_0$ fulfilling $2k+1\le \kappa$ by mathematical induction. $\square$

In view of Definition 4.7 and Notation 4.10, we obtain the following result:

Lemma 11.11

Suppose $\kappa \ge 1$. Let $(s_j)_{j=0}^{\kappa }\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,\kappa ,\alpha }$, let $k\in \mathbb {N}_0$ be such that $2k+1\le \kappa$, let $\left( {\eta };{\theta }\right) \in \hat{\mathcal {P}}\left[ {\mathfrak {h}_{2k}} \right]$, and let $\mathcal {E}\in \mathscr {E}\left( {\eta ,\theta }\right)$. Then:

• (a) For all $z\in \mathbb {C}\backslash \left( {\mathbb {R}\cup \mathcal {E}}\right)$, the pair $\left( {\eta };{\theta }\right)$ fulfills

  $$\begin{aligned} \det \left( {\textbf{b}_{k}\left( {z}\right) \mathfrak {h}_{2k}^{\mathord {+}}\eta \left( {z}\right) +\textbf{b}_{k+1}\left( {z}\right) \theta \left( {z}\right) }\right)&\ne 0 \end{aligned}$$

  (11.12)

  and

  $$\begin{aligned} \det \left( {\mathfrak {b}_{k}\left( {z}\right) \mathfrak {h}_{2k}^{\mathord {+}}\eta \left( {z}\right) +\mathfrak {b}_{k+1}\left( {z}\right) \theta \left( {z}\right) }\right)&\ne 0. \end{aligned}$$

  (11.13)

• (b) For all $w\in \mathbb {C}\backslash \left( {\mathbb {R}\cup \mathcal {E}^{\mathord {\vee }}}\right)$, the inequalities $\det \left( {\left[ {\eta \left( {\overline{w}}\right) } \right] ^*\mathfrak {h}_{2k}^{\mathord {+}}\textbf{d}_{k}\left( {w}\right) +\left[ {\theta \left( {\overline{w}}\right) } \right] ^*\textbf{d}_{k+1}\left( {w}\right) }\right) \ne 0$ and $\det \left( {\left[ {\eta \left( {\overline{w}}\right) } \right] ^*\mathfrak {h}_{2k}^{\mathord {+}}\mathfrak {d}_{k}\left( {w}\right) +\left[ {\theta \left( {\overline{w}}\right) } \right] ^*\mathfrak {d}_{k+1}\left( {w}\right) }\right) \ne 0$ hold true.

Proof

First observe that Remark 7.6 provides $(s_j)_{j=0}^{\kappa }\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,\kappa }$.

(a) We consider an arbitrary $z\in \mathbb {C}\backslash \left( {\mathbb {R}\cup \mathcal {E}}\right)$. According to Lemma 6.5, we have (6.2). In particular, $\mathfrak {h}_{2k}\in \mathbb {C}_\succcurlyeq ^{{q\times q}}$. By virtue Notation 4.10, Definition 4.7, Remark 4.9(a), and Remark A.10, we infer that $\left( {\eta };{\theta }\right)$ belongs to $\mathcal{P}\mathcal{R}_{q}\left( {\mathbb {C}\backslash \mathbb {R}}\right)$ and fulfills ${{\,\textrm{rank}\,}}\bigl [{\begin{matrix}\eta \left( {z}\right) \\ \theta \left( {z}\right) \end{matrix}}\bigr ]=q$ and $\left( {\Im z}\right) ^{-1}\Im \left( {\left[ {\theta \left( {z}\right) } \right] ^*\eta \left( {z}\right) }\right) \in \mathbb {C}_\succcurlyeq ^{{q\times q}}$ as well as $\mathfrak {h}_{2k}\mathfrak {h}_{2k}^{\mathord {+}}\eta \left( {z}\right) =\eta \left( {z}\right)$. Now we are going to prove (11.12). Setting

$$\begin{aligned} B_{k}:=\textbf{b}_{k}\left( {z}\right) ,\, D_{k}:=\textbf{d}_{k}\left( {z}\right) ,\, B_{k+1}:=\textbf{b}_{k+1}\left( {z}\right) , \text { and } D_{k+1}:=\textbf{d}_{k+1}\left( {z}\right) ,\qquad \end{aligned}$$

(11.14)

Lemma 11.8 yields

$$\begin{aligned} \det B_{k}&\ne 0,&\det D_{k}&\ne 0,&\det B_{k+1}&\ne 0,&\det D_{k+1}&\ne 0, \end{aligned}$$

(11.15)

and

$$\begin{aligned} \mathfrak {h}_{2k} B_{k}^{-1}B_{k+1} = D_{k+1} D_{k}^{-1}\mathfrak {h}_{2k}, \end{aligned}$$

(11.16)

whereas Lemma 11.10 provides

$$\begin{aligned} \chi _{2k+1}\left( {z}\right) =\mathfrak {h}_{2k} B_{k}^{-1}B_{k+1}. \end{aligned}$$

(11.17)

We consider an arbitrary $v\in \mathcal {N}\left( { B_{k}\mathfrak {h}_{2k}^{\mathord {+}}\eta \left( {z}\right) + B_{k+1}\theta \left( {z}\right) }\right)$. Then

$$\begin{aligned} B_{k}\mathfrak {h}_{2k}^{\mathord {+}}\eta \left( {z}\right) v+ B_{k+1}\theta \left( {z}\right) v =O_{{q\times 1}}. \end{aligned}$$

(11.18)

Regarding (11.15), we can multiply (11.18) from the left by $B_{k}^{-1}$ to obtain

$$\begin{aligned} \mathfrak {h}_{2k}^{\mathord {+}}\eta \left( {z}\right) v+ B_{k}^{-1}B_{k+1}\theta \left( {z}\right) v =O_{{q\times 1}}. \end{aligned}$$

(11.19)

Using (11.19) and $\mathfrak {h}_{2k}\mathfrak {h}_{2k}^{\mathord {+}}\eta \left( {z}\right) =\eta \left( {z}\right)$, we get

$$\begin{aligned} \begin{aligned} O_{{q\times 1}}&=\mathfrak {h}_{2k}\left[ {\mathfrak {h}_{2k}^{\mathord {+}}\eta \left( {z}\right) v+ B_{k}^{-1}B_{k+1}\theta \left( {z}\right) v} \right] \\&=\eta \left( {z}\right) v+\mathfrak {h}_{2k} B_{k}^{-1}B_{k+1}\theta \left( {z}\right) v. \end{aligned}\end{aligned}$$

(11.20)

Multiplying both sides of (11.20) from the left by $v^*\left[ {\theta \left( {z}\right) } \right] ^*$ and using (11.17), we obtain

$$\begin{aligned} \begin{aligned} 0&=v^*\left[ {\theta \left( {z}\right) } \right] ^*\eta \left( {z}\right) v+v^*\left[ {\theta \left( {z}\right) } \right] ^*\mathfrak {h}_{2k} B_{n}^{-1}B_{n+1}\theta \left( {z}\right) v\\&=v^*\left[ {\theta \left( {z}\right) } \right] ^*\eta \left( {z}\right) v+\left[ {\theta \left( {z}\right) v} \right] ^*\chi _{2k+1}\left( {z}\right) \left[ {\theta \left( {z}\right) v} \right] . \end{aligned}\end{aligned}$$

(11.21)

In view of $z\in \mathbb {C}\backslash \mathbb {R}$, multiplying both sides of (11.21) by $\left( {\Im z}\right) ^{-1}$, taking the imaginary part, and regarding Remarks A.1 and A.2 as well as (6.2) and $\left( {\Im z}\right) ^{-1}\Im \left( {\left[ {\theta \left( {z}\right) } \right] ^*\eta \left( {z}\right) }\right) \in \mathbb {C}_\succcurlyeq ^{{q\times q}}$, we conclude

$$\begin{aligned}&0 =\Im \left( {v^*\left( {\left( {\Im z}\right) ^{-1}\left[ {\theta \left( {z}\right) } \right] ^*\eta \left( {z}\right) }\right) v+\left[ {\theta \left( {z}\right) v} \right] ^*\left[ {\left( {\Im z}\right) ^{-1}\chi _{2k+1}\left( {z}\right) } \right] \left[ {\theta \left( {z}\right) v} \right] }\right) \nonumber \\&=v^*\left[ {\left( {\Im z}\right) ^{-1}\Im \left( {\left[ {\theta \left( {z}\right) } \right] ^*\eta \left( {z}\right) }\right) } \right] v+\left[ {\theta \left( {z}\right) v} \right] ^*\left[ {\left( {\Im z}\right) ^{-1}\Im \chi _{2k+1}\left( {z}\right) } \right] \left[ {\theta \left( {z}\right) v} \right] \nonumber \\&\ge v^*\left[ {\left( {\Im z}\right) ^{-1}\Im \left( {\left[ {\theta \left( {z}\right) } \right] ^*\eta \left( {z}\right) }\right) } \right] v+\left[ {\theta \left( {z}\right) v} \right] ^*\mathfrak {h}_{2 k }\left[ {\theta \left( {z}\right) v} \right] \nonumber \\&\ge \left[ {\theta \left( {z}\right) v} \right] ^*\mathfrak {h}_{2 k }\left[ {\theta \left( {z}\right) v} \right] \ge 0. \end{aligned}$$

(11.22)

Consequently, we get $\left[ {\theta \left( {z}\right) v} \right] ^*\mathfrak {h}_{2 k }\left[ {\theta \left( {z}\right) v} \right] =0$ and, because of $\mathfrak {h}_{2k}\in \mathbb {C}_\succcurlyeq ^{{q\times q}}$, then $\mathfrak {h}_{2k}\theta \left( {z}\right) v=O_{{q\times 1}}$ follows. Using additionally (11.20) and (11.16), we conclude

$$\begin{aligned} O_{{q\times 1}}&=\eta \left( {z}\right) v+\mathfrak {h}_{2 k } B_{k}^{-1}B_{ k +1}\theta \left( {z}\right) v \nonumber \\&=\eta \left( {z}\right) v+ D_{ k +1} D_{k}^{-1}\mathfrak {h}_{2 k }\theta \left( {z}\right) v =\eta \left( {z}\right) v. \end{aligned}$$

(11.23)

Combining (11.18) and (11.23), we obtain $B_{k+1}\theta \left( {z}\right) v=O_{{q\times 1}}$ and, thus, (11.15) justifies that $\theta \left( {z}\right) v=O_{{q\times 1}}$. Taking additionally into account (11.23) and regarding ${{\,\textrm{rank}\,}}\bigl [{\begin{matrix}\eta \left( {z}\right) \\ \theta \left( {z}\right) \end{matrix}}\bigr ]=q$, we infer $v=O_{{q\times 1}}$. Hence,

$$\begin{aligned} \mathcal {N}\left( { B_{k}\mathfrak {h}_{2k}^{\mathord {+}}\eta \left( {z}\right) + B_{k+1}\theta \left( {z}\right) }\right) \subseteq \left\{ {O_{{q\times 1}}} \right\} . \end{aligned}$$

(11.24)

In view of (11.14), therefore (11.12) is checked. In order to prove (11.13), we now set

$$\begin{aligned} B_{k}:=\mathfrak {b}_{k}\left( {z}\right) ,\; D_{k}:=\mathfrak {d}_{k}\left( {z}\right) ,\; B_{ k +1}:=\mathfrak {b}_{ k +1}\left( {z}\right) , \text { and } D_{ k +1}:=\mathfrak {d}_{ k +1}\left( {z}\right) . \end{aligned}$$

(11.25)

Regarding $(s_j)_{j=0}^{\kappa }\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,\kappa }$, we can then use Lemma 5.9 to obtain (11.15) and Remark 6.2 to get (11.17) and (11.16). We consider an arbitrary $v\in \mathcal {N}\left( { B_{k}\mathfrak {h}_{2k}^{\mathord {+}}\eta \left( {z}\right) + B_{k+1}\theta \left( {z}\right) }\right)$. Then (11.18) is fulfilled. Regarding (11.15), we can multiply (11.18) from the left by $B_{k}^{-1}$ to obtain (11.19). Using (11.19) and $\mathfrak {h}_{2k}\mathfrak {h}_{2k}^{\mathord {+}}\eta \left( {z}\right) =\eta \left( {z}\right)$, we again get (11.20). Multiplying both sides of (11.20) from the left by $v^*\left[ {\theta \left( {z}\right) } \right] ^*$ and using (11.17), we obtain (11.21). In view of $z\in \mathbb {C}\backslash \mathbb {R}$, multiplying both sides of (11.21) by $\left( {\Im z}\right) ^{-1}$, taking the imaginary part and regarding Remarks A.1 and A.2 as well as (6.2) and $\left( {\Im z}\right) ^{-1}\Im \left( {\left[ {\theta \left( {z}\right) } \right] ^*\eta \left( {z}\right) }\right) \in \mathbb {C}_\succcurlyeq ^{{q\times q}}$, we conclude that (11.22) is valid. Consequently, we get $\left[ {\theta \left( {z}\right) v} \right] ^*\mathfrak {h}_{2 k }\left[ {\theta \left( {z}\right) v} \right] =0$ and, because of $\mathfrak {h}_{2k}\in \mathbb {C}_\succcurlyeq ^{{q\times q}}$, then $\mathfrak {h}_{2k}\theta \left( {z}\right) v=O_{{q\times 1}}$ follows. Using additionally (11.20) and (11.16), we conclude (11.23). Combining (11.18) and (11.23), we obtain $B_{k+1}\theta \left( {z}\right) v=O_{{q\times 1}}$ and, thus, (11.15) justifies that $\theta \left( {z}\right) v=O_{{q\times 1}}$. Taking additionally into account (11.23) and regarding ${{\,\textrm{rank}\,}}\bigl [{\begin{matrix}\eta \left( {z}\right) \\ \theta \left( {z}\right) \end{matrix}}\bigr ]=q$, we infer $v=O_{{q\times 1}}$. Consequently, (11.24) is checked. Taking into account (11.25), we see then that (11.13) is valid.

(b) Regarding $(s_j)_{j=0}^{\kappa }\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,\kappa }$, Remark 5.3 yields $\mathfrak {h}_{2k}^*=\mathfrak {h}_{2k}$. Thus, using Remark A.12, we can infer $\left( {\mathfrak {h}_{2k}^{\mathord {+}}}\right) ^*=\mathfrak {h}_{2k}^{\mathord {+}}$. We consider an arbitrary $w\in \mathbb {C}\backslash \left( {\mathbb {R}\cup \mathcal {E}^{\mathord {\vee }}}\right)$. In view of (2.1), then $\overline{w}\in \mathbb {C}\backslash \left( {\mathbb {R}\cup \mathcal {E}}\right)$. Since Remark 9.3 shows that $(s_j)_{j=0}^{\kappa }$ is a sequence of Hermitian matrices, we can use Remark 11.2 to obtain furthermore $\left[ {\textbf{b}_{j}\left( {\overline{w}}\right) } \right] ^*=\textbf{d}_{j}\left( {w}\right)$ for each $j\in \left\{ {k,k+1} \right\}$. Consequently, we get $\left[ {\textbf{b}_{k}\left( {\overline{w}}\right) \mathfrak {h}_{2k}^{\mathord {+}}\eta \left( {\overline{w}}\right) +\textbf{b}_{k+1}\left( {\overline{w}}\right) \theta \left( {\overline{w}}\right) } \right] ^*=\left[ {\eta \left( {\overline{w}}\right) } \right] ^*\mathfrak {h}_{2k}^{\mathord {+}}\textbf{d}_{k}\left( {w}\right) +\left[ {\theta \left( {\overline{w}}\right) } \right] ^*\textbf{d}_{k+1}\left( {w}\right)$. Since part (a) implies $\det \left( {\textbf{b}_{k}\left( {\overline{w}}\right) \mathfrak {h}_{2k}^{\mathord {+}}\eta \left( {\overline{w}}\right) +\textbf{b}_{k+1}\left( {\overline{w}}\right) \theta \left( {\overline{w}}\right) }\right) \ne 0$, then $\det \left( {\left[ {\eta \left( {\overline{w}}\right) } \right] ^*\mathfrak {h}_{2k}^{\mathord {+}}\textbf{d}_{k}\left( {w}\right) +\left[ {\theta \left( {\overline{w}}\right) } \right] ^*\textbf{d}_{k+1}\left( {w}\right) }\right) \ne 0$ follows. Using Remark 5.8 instead of Remark 11.2, we can infer analogously $\det \left( {\left[ {\eta \left( {\overline{w}}\right) } \right] ^*\mathfrak {h}_{2k}^{\mathord {+}}\mathfrak {d}_{k}\left( {w}\right) +\left[ {\theta \left( {\overline{w}}\right) } \right] ^*\mathfrak {d}_{k+1}\left( {w}\right) }\right) \ne 0$. $\square$

Lemma 11.12

Suppose $\kappa \ge 1$. Let $(s_j)_{j=0}^{\kappa }\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,\kappa ,\alpha }$. For all $k \in \mathbb {N}_0$ such that $2 k +1\le \kappa$, then

$$\begin{aligned} \begin{bmatrix}\mathfrak {c}_{k}&{}\mathfrak {d}_{k}\\ \mathfrak {c}_{ k +1}&{}\mathfrak {d}_{ k +1}\end{bmatrix} \left( {-\textrm{i}\tilde{J}_{q}}\right) \begin{bmatrix}\textbf{a}_{k}&{}\textbf{a}_{ k +1}\\ \textbf{b}_{k}&{}\textbf{b}_{ k +1}\end{bmatrix}&=\begin{bmatrix} O_{{q\times q}}&{}\mathfrak {h}_{2 k }\\ -\mathfrak {h}_{2 k }&{}O_{{q\times q}}\end{bmatrix}, \end{aligned}$$

(11.26)

$$\begin{aligned} \begin{bmatrix}\textbf{c}_{k}&{}\textbf{d}_{k}\\ \textbf{c}_{ k +1}&{}\textbf{d}_{ k +1}\end{bmatrix} \left( {-\textrm{i}\tilde{J}_{q}}\right) \begin{bmatrix}\mathfrak {a}_{k}&{}\mathfrak {a}_{ k +1}\\ \mathfrak {b}_{k}&{}\mathfrak {b}_{ k +1}\end{bmatrix}&=\begin{bmatrix} O_{{q\times q}}&{}\mathfrak {h}_{2 k }\\ -\mathfrak {h}_{2 k }&{}O_{{q\times q}}\end{bmatrix}, \end{aligned}$$

(11.27)

$$\begin{aligned} \begin{bmatrix}\mathfrak {c}_{k}&{}\mathfrak {d}_{k}\\ \mathfrak {c}_{ k +1}&{}\mathfrak {d}_{ k +1}\end{bmatrix} \left( {-\textrm{i}\tilde{J}_{q}}\right) \begin{bmatrix}\mathfrak {a}_{k}&{}\mathfrak {a}_{ k +1}\\ \mathfrak {b}_{k}&{}\mathfrak {b}_{ k +1}\end{bmatrix}&=\begin{bmatrix} O_{{q\times q}}&{}\mathfrak {h}_{2 k }\\ -\mathfrak {h}_{2 k }&{}O_{{q\times q}}\end{bmatrix}, \end{aligned}$$

(11.28)

and

$$\begin{aligned} \begin{bmatrix}\textbf{c}_{k}&{}\textbf{d}_{k}\\ \textbf{c}_{ k +1}&{}\textbf{d}_{ k +1}\end{bmatrix} \left( {-\textrm{i}\tilde{J}_{q}}\right) \begin{bmatrix}\textbf{a}_{k}&{}\textbf{a}_{ k +1}\\ \textbf{b}_{k}&{}\textbf{b}_{ k +1}\end{bmatrix}&=\begin{bmatrix} O_{{q\times q}}&{}\mathfrak {h}_{2 k }\\ -\mathfrak {h}_{2 k }&{}O_{{q\times q}}\end{bmatrix}. \end{aligned}$$

(11.29)

Proof

Remark 9.2 yields (9.1), whereas Lemma 9.4 provides (9.5). According to Remark 9.3, we have furthermore (9.3). Remark 7.6 yields $(s_j)_{j=0}^{\kappa }\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,\kappa }$. Thus, Remark 5.3 provides (5.4) and (5.5). We proceed by mathematical induction. Using (4.1), (5.6), (5.7), Notation 5.5, Lemmata 11.5 and 11.6, (9.1), (9.5), and (9.3), we can infer

and

Analogously, using (4.1), (5.6), (5.7), and (5.4), we can conclude that (11.28) is valid for $k=0$, and, using (4.1), Lemmata 11.5 and 11.6, (9.3), and (9.1), we get furthermore that (11.29) holds true for $k=0$. Now suppose $\kappa \ge 3$. Then, we have already shown that there exists an integer $\ell \in \mathbb {N}$ fulfilling $2\ell +1\le \kappa$ such that (11.26)–(11.29) hold true for $k=\ell -1$. Using Remark 5.6 and (11.26) for $k=\ell -1$, we get

Taking into account Remark 11.7, (9.1), and (5.5), then

follows, where

$$\begin{aligned}{} & {} R :=\left( {\epsilon I_{q}-\mathfrak {h}_{2\ell +1}\mathfrak {h}_{2\ell }^{\mathord {+}}}\right) \mathfrak {h}_{2\ell -2}\mathfrak {k}_{2\ell -2}^{\mathord {+}}\mathfrak {k}_{2\ell }\\{} & {} \quad -\mathfrak {h}_{2\ell }\mathfrak {h}_{2\ell -2}^{\mathord {+}}\mathfrak {h}_{2\ell -2}\left[ {\left( {\epsilon -\alpha }\right) I_{q}-\left( {\mathfrak {k}_{2\ell -1}^{\mathord {+}}\mathfrak {k}_{2\ell }+\mathfrak {k}_{2\ell }^{\mathord {+}}\mathfrak {k}_{2\ell +1}}\right) } \right] . \end{aligned}$$

Using (9.1), (5.5), (5.4), (9.3), and (9.5), we can further conclude

$$\begin{aligned} \begin{aligned} R&=\left( {\epsilon I_{q}-\mathfrak {h}_{2\ell +1}\mathfrak {h}_{2\ell }^{\mathord {+}}}\right) \mathfrak {h}_{2\ell -2}\mathfrak {h}_{2\ell -2}^{\mathord {+}}\mathfrak {h}_{2\ell }\\&\qquad -\mathfrak {k}_{2\ell }\mathfrak {k}_{2\ell -2}^{\mathord {+}}\mathfrak {k}_{2\ell -2}\left[ {\left( {\epsilon -\alpha }\right) I_{q}-\left( {\mathfrak {k}_{2\ell -1}^{\mathord {+}}\mathfrak {k}_{2\ell }+\mathfrak {k}_{2\ell }^{\mathord {+}}\mathfrak {k}_{2\ell +1}}\right) } \right] \\&=\left( {\epsilon I_{q}-\mathfrak {h}_{2\ell +1}\mathfrak {h}_{2\ell }^{\mathord {+}}}\right) \mathfrak {h}_{2\ell }-\mathfrak {k}_{2\ell }\left[ {\left( {\epsilon -\alpha }\right) I_{q}-\left( {\mathfrak {k}_{2\ell -1}^{\mathord {+}}\mathfrak {k}_{2\ell }+\mathfrak {k}_{2\ell }^{\mathord {+}}\mathfrak {k}_{2\ell +1}}\right) } \right] \\&=\epsilon \mathfrak {h}_{2\ell }-\mathfrak {h}_{2\ell +1}\mathfrak {h}_{2\ell }^{\mathord {+}}\mathfrak {h}_{2\ell }-\epsilon \mathfrak {k}_{2\ell }+\alpha \mathfrak {k}_{2\ell }+\mathfrak {k}_{2\ell }\mathfrak {k}_{2\ell -1}^{\mathord {+}}\mathfrak {k}_{2\ell }+\mathfrak {k}_{2\ell }\mathfrak {k}_{2\ell }^{\mathord {+}}\mathfrak {k}_{2\ell +1}\\&=\epsilon \mathfrak {h}_{2\ell }-\mathfrak {h}_{2\ell +1}-\epsilon \mathfrak {k}_{2\ell }+\alpha \mathfrak {k}_{2\ell }+\mathfrak {k}_{2\ell }\mathfrak {k}_{2\ell -1}^{\mathord {+}}\mathfrak {k}_{2\ell }+\mathfrak {k}_{2\ell +1}\\&=\epsilon \mathfrak {k}_{2\ell }-\mathfrak {h}_{2\ell +1}-\epsilon \mathfrak {k}_{2\ell }+\mathfrak {h}_{2\ell +1} =O_{{q\times q}}. \end{aligned} \end{aligned}$$

Consequently, (11.26) holds true for $k=\ell$. Analogously, one can check that (11.27), (11.28), and (11.29) are valid for $k=\ell$ as well. Thus, (11.26)–(11.29) are inductively proved for all $k\in \mathbb {N}_0$ fulfilling $2k+1\le \kappa$. $\square$

Proposition 11.13

Suppose $\kappa \ge 1$. Let $(s_j)_{j=0}^{\kappa }\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,\kappa ,\alpha }$, let $k \in \mathbb {N}_0$ be such that $2 k +1\le \kappa$, and let $\left( {\eta };{\theta }\right) \in \hat{\mathcal {P}}\left[ {\mathfrak {h}_{2 k }} \right]$. Let $\mathcal {E}\in \mathscr {E}\left( {\eta ,\theta }\right)$ and let $\hat{\mathcal {E}}:=\mathcal {E}\cup \mathcal {E}^{\mathord {\vee }}$. For all $z\in \mathbb {C}\backslash \left( {\mathbb {R}\cup \hat{\mathcal {E}}}\right)$, then (11.13), (11.12),

$$\begin{aligned} \det \left( {\left[ {\eta \left( {\overline{z}}\right) } \right] ^*\mathfrak {h}_{2 k }^{\mathord {+}}\mathfrak {d}_{k}\left( {z}\right) +\left[ {\theta \left( {\overline{z}}\right) } \right] ^*\mathfrak {d}_{ k +1}\left( {z}\right) }\right)&\ne 0, \end{aligned}$$

(11.30)

and

$$\begin{aligned} \det \left( {\left[ {\eta \left( {\overline{z}}\right) } \right] ^*\mathfrak {h}_{2 k }^{\mathord {+}}\textbf{d}_{k}\left( {z}\right) +\left[ {\theta \left( {\overline{z}}\right) } \right] ^*\textbf{d}_{ k +1}\left( {z}\right) }\right)&\ne 0 \end{aligned}$$

(11.31)

hold true and the matrix-valued function $F:=-\left( {\mathfrak {a}_{k}\mathfrak {h}_{2 k }^{\mathord {+}}\eta +\mathfrak {a}_{ k +1}\theta }\right) \left( {\mathfrak {b}_{k}\mathfrak {h}_{2 k }^{\mathord {+}}\eta +\mathfrak {b}_{ k +1}\theta }\right) ^{-1}$ admits, for all $z\in \mathbb {C}\backslash \left( {\mathbb {R}\cup \hat{\mathcal {E}}}\right)$, the representations

and

Proof

Let $z\in \mathbb {C}\backslash \left( {\mathbb {R}\cup \hat{\mathcal {E}}}\right)$. Then (11.13), (11.12), (11.30), and (11.31) follow immediately from Lemma 11.11. Since Remark 7.6 provides $(s_j)_{j=0}^{\kappa }\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,\kappa }$, Remark 5.3 yields $\mathfrak {h}_{2k}^*=\mathfrak {h}_{2k}$. Thus, using Remark A.12, we can infer $\left( {\mathfrak {h}_{2k}^{\mathord {+}}}\right) ^*=\mathfrak {h}_{2k}^{\mathord {+}}$. According to Notation 4.10, the pair $\left( {\eta };{\theta }\right)$ belongs to $\mathcal{P}\mathcal{R}_{q}\left( {\mathbb {C}\backslash \mathbb {R}}\right)$ and fulfills $\mathbb {P}_{\mathcal {R}\left( {\mathfrak {h}_{2k}}\right) }\eta =\eta$. Remark 4.9(b) then yields $\left[ {\eta \left( {\overline{z}}\right) } \right] ^*\theta \left( {z}\right) =\left[ {\theta \left( {\overline{z}}\right) } \right] ^*\eta \left( {z}\right)$, whereas Remark A.10 provides $\mathfrak {h}_{2k}\mathfrak {h}_{2k}^{\mathord {+}}\eta =\eta$. Consequently, we obtain

(11.32)

Let

$$\begin{aligned}{\mathfrak {A}}&:=\mathfrak {a}_{k}\left( {z}\right) \mathfrak {h}_{2k}^{\mathord {+}}\eta \left( {z}\right) +\mathfrak {a}_{k+1}\left( {z}\right) \theta \left( {z}\right) ,&{\mathfrak {C}}&:=\left[ {\eta \left( {\overline{z}}\right) } \right] ^*\mathfrak {h}_{2k}^{\mathord {+}}\mathfrak {c}_{k}\left( {z}\right) +\left[ {\theta \left( {\overline{z}}\right) } \right] ^*\mathfrak {c}_{k+1}\left( {z}\right) ,\\ {\mathfrak {B}}&:=\mathfrak {b}_{k}\left( {z}\right) \mathfrak {h}_{2k}^{\mathord {+}}\eta \left( {z}\right) +\mathfrak {b}_{k+1}\left( {z}\right) \theta \left( {z}\right) ,&{\mathfrak {D}}&:=\left[ {\eta \left( {\overline{z}}\right) } \right] ^*\mathfrak {h}_{2k}^{\mathord {+}}\mathfrak {d}_{k}\left( {z}\right) +\left[ {\theta \left( {\overline{z}}\right) } \right] ^*\mathfrak {d}_{k+1}\left( {z}\right) \end{aligned}$$

as well as

$$\begin{aligned} {\textbf{A}}&:=\textbf{a}_{k}\left( {z}\right) \mathfrak {h}_{2k}^{\mathord {+}}\eta \left( {z}\right) +\textbf{a}_{k+1}\left( {z}\right) \theta \left( {z}\right) ,&{\textbf{C}}&:=\left[ {\eta \left( {\overline{z}}\right) } \right] ^*\mathfrak {h}_{2k}^{\mathord {+}}\textbf{c}_{k}\left( {z}\right) +\left[ {\theta \left( {\overline{z}}\right) } \right] ^*\textbf{c}_{k+1}\left( {z}\right) ,\\ {\textbf{B}}&:=\textbf{b}_{k}\left( {z}\right) \mathfrak {h}_{2k}^{\mathord {+}}\eta \left( {z}\right) +\textbf{b}_{k+1}\left( {z}\right) \theta \left( {z}\right) ,&{\textbf{D}}&:=\left[ {\eta \left( {\overline{z}}\right) } \right] ^*\mathfrak {h}_{2k}^{\mathord {+}}\textbf{d}_{k}\left( {z}\right) +\left[ {\theta \left( {\overline{z}}\right) } \right] ^*\textbf{d}_{k+1}\left( {z}\right) . \end{aligned}$$

From Lemma 11.12 we have (11.26)–(11.29). By virtue of (4.1), (11.28), and (11.32), we get

Taking additionally into account (11.13) and (11.30), then ${\mathfrak {A}}{\mathfrak {B}}^{-1}={\mathfrak {D}}^{-1}{\mathfrak {C}}$ follows. Similarly, (4.1), (11.29), and (11.32) yield ${\textbf{D}}{\textbf{A}}-{\textbf{C}}{\textbf{B}}=O_{{q\times q}}$, which, in view of (11.12) and (11.31), implies ${\textbf{A}}{\textbf{B}}^{-1}={\textbf{D}}^{-1}{\textbf{C}}$. Using (4.1), (11.26), (11.32), (11.12), and (11.30), we obtain analogously ${\textbf{A}}{\textbf{B}}^{-1}={\mathfrak {D}}^{-1}{\mathfrak {C}}$. Finally, summarizing, we infer ${\mathfrak {A}}{\mathfrak {B}}^{-1}={\mathfrak {D}}^{-1}{\mathfrak {C}}={\textbf{A}}{\textbf{B}}^{-1}={\textbf{D}}^{-1}{\textbf{C}}$. $\square$

Corollary 11.14

Suppose $\kappa \ge 1$. Let $(s_j)_{j=0}^{\kappa }\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,\kappa ,\alpha }$, let $k\in \mathbb {N}_0$ be such that $2k+1\le \kappa$, and let $\left( {\phi };{\psi }\right) \in \mathcal {P}\left[ {\mathfrak {h}_{2k}} \right]$ and $\mathcal {D}\in \mathscr {D}\left( {\phi ,\psi }\right)$. For all $z\in \Pi _{\mathord {+}}\backslash \mathcal {D}$, then $\det \left( {\mathfrak {b}_{k}\left( {z}\right) \mathfrak {h}_{2k}^{\mathord {+}}\phi \left( {z}\right) +\mathfrak {b}_{k+1}\left( {z}\right) \psi \left( {z}\right) }\right) \ne 0$ and $\det \left( {\textbf{b}_{k}\left( {z}\right) \mathfrak {h}_{2k}^{\mathord {+}}\phi \left( {z}\right) +\textbf{b}_{k+1}\left( {z}\right) \psi \left( {z}\right) }\right) \ne 0$ as well as

$$\begin{aligned}{} & {} -\left[ {\mathfrak {a}_{k}\left( {z}\right) \mathfrak {h}_{2k}^{\mathord {+}}\phi \left( {z}\right) +\mathfrak {a}_{k+1}\left( {z}\right) \psi \left( {z}\right) } \right] \left[ {\mathfrak {b}_{k}\left( {z}\right) \mathfrak {h}_{2k}^{\mathord {+}}\phi \left( {z}\right) +\mathfrak {b}_{k+1}\left( {z}\right) \psi \left( {z}\right) } \right] ^{-1}\\{} & {} \quad =-\left[ {\textbf{a}_{k}\left( {z}\right) \mathfrak {h}_{2k}^{\mathord {+}}\phi \left( {z}\right) +\textbf{a}_{k+1}\left( {z}\right) \psi \left( {z}\right) } \right] \left[ {\textbf{b}_{k}\left( {z}\right) \mathfrak {h}_{2k}^{\mathord {+}}\phi \left( {z}\right) +\textbf{b}_{k+1}\left( {z}\right) \psi \left( {z}\right) } \right] ^{-1}. \end{aligned}$$

Proof

According to Notation 4.5, we have $\left( {\phi };{\psi }\right) \in \mathcal{P}\mathcal{R}_{q}\left( {\Pi _{\mathord {+}}}\right)$. Thus, Lemma 4.11(e) shows that $\left( {\phi _{\mathord {\diamond }}};{\psi _{\mathord {\diamond }}}\right)$ given by (4.4) belongs to $\mathcal{P}\mathcal{R}_{q}\left( {\mathbb {C}\backslash \mathbb {R}}\right)$ and that $\mathcal {E}:=\mathcal {D}\cup \mathcal {D}^{\mathord {\vee }}$ belongs to $\mathscr {E}\left( {\phi _{\mathord {\diamond }},\psi _{\mathord {\diamond }}}\right)$. We now consider an arbitrary $z\in \Pi _{\mathord {+}}\backslash \mathcal {D}$. In view of Definition 4.2, we have $\mathcal {D}\subseteq \Pi _{\mathord {+}}$. Regarding (2.1), thus $z\notin \mathcal {D}^{\mathord {\vee }}$. Consequently, $z\notin \mathcal {E}$. By virtue of (2.1), it is readily checked that $\mathcal {E}^{\mathord {\vee }}=\mathcal {E}$. Hence, $\hat{\mathcal {E}}:=\mathcal {E}\cup \mathcal {E}^{\mathord {\vee }}$ fulfills $\hat{\mathcal {E}}=\mathcal {E}$. Summarizing, we can conclude $z\in \mathbb {C}\backslash \left( {\mathbb {R}\cup \hat{\mathcal {E}}}\right)$. Since Remark 7.6 provides $(s_j)_{j=0}^{\kappa }\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,\kappa }$, we can apply Remark 5.3 to obtain $\mathfrak {h}_{2k}\in \mathbb {C}_\textrm{H}^{{q\times q}}$. From Lemma 4.12 we can infer then $\left( {\phi _{\mathord {\diamond }}};{\psi _{\mathord {\diamond }}}\right) \in \hat{\mathcal {P}}\left[ {\mathfrak {h}_{2k}} \right]$. Taking additionally into account $z\in \mathbb {C}\backslash \left( {\mathbb {R}\cup \hat{\mathcal {E}}}\right)$, the application of Proposition 11.13 then yields $\det \left( {\mathfrak {b}_{k}\left( {z}\right) \mathfrak {h}_{2k}^{\mathord {+}}\phi _{\mathord {\diamond }}\left( {z}\right) +\mathfrak {b}_{k+1}\left( {z}\right) \psi _{\mathord {\diamond }}\left( {z}\right) }\right) \ne 0$ and $\det \left( {\textbf{b}_{k}\left( {z}\right) \mathfrak {h}_{2k}^{\mathord {+}}\phi _{\mathord {\diamond }}\left( {z}\right) +\textbf{b}_{k+1}\left( {z}\right) \psi _{\mathord {\diamond }}\left( {z}\right) }\right) \ne 0$ and that $F:=-\left( {\mathfrak {a}_{k}\mathfrak {h}_{2 k }^{\mathord {+}}\phi _{\mathord {\diamond }}+\mathfrak {a}_{ k +1}\psi _{\mathord {\diamond }}}\right) \left( {\mathfrak {b}_{k}\mathfrak {h}_{2 k }^{\mathord {+}}\phi _{\mathord {\diamond }}+\mathfrak {b}_{ k +1}\psi _{\mathord {\diamond }}}\right) ^{-1}$ fulfills $F\left( {z}\right) =-\left[ {\mathfrak {a}_{k}\left( {z}\right) \mathfrak {h}_{2k}^{\mathord {+}}\phi _{\mathord {\diamond }}\left( {z}\right) +\mathfrak {a}_{k+1}\left( {z}\right) \psi _{\mathord {\diamond }}\left( {z}\right) } \right] \left[ {\mathfrak {b}_{k}\left( {z}\right) \mathfrak {h}_{2k}^{\mathord {+}}\phi _{\mathord {\diamond }}\left( {z}\right) +\mathfrak {b}_{k+1}\left( {z}\right) \psi _{\mathord {\diamond }}\left( {z}\right) } \right] ^{-1}$ and $F\left( {z}\right) =-\left[ {\textbf{a}_{k}\left( {z}\right) \mathfrak {h}_{2k}^{\mathord {+}}\phi _{\mathord {\diamond }}\left( {z}\right) +\textbf{a}_{k+1}\left( {z}\right) \psi _{\mathord {\diamond }}\left( {z}\right) } \right] \left[ {\textbf{b}_{k}\left( {z}\right) \mathfrak {h}_{2k}^{\mathord {+}}\phi _{\mathord {\diamond }}\left( {z}\right) +\textbf{b}_{k+1}\left( {z}\right) \psi _{\mathord {\diamond }}\left( {z}\right) } \right] ^{-1}$. Since Lemma 4.11(d) provides $\phi _{\mathord {\diamond }}\left( {z}\right) =\phi \left( {z}\right)$ and $\psi _{\mathord {\diamond }}\left( {z}\right) =\psi \left( {z}\right)$, the assertions follow. $\square$

12 The Second ${[\alpha ,\infty )}$-quadruple of Matrix Polynomials

Analogous to Sect. 11, we now focus our attention to another subsystem of the quadruple considered in Sect. 10.

Remark 12.1

(cf. [13, Notation 14.4]) Suppose $\kappa \ge 1$. Let $(s_j)_{j=0}^{\kappa }$ be a sequence of complex ${q\times q}$ matrices. In view of Remark 10.8, for each $k\in \mathbb {N}_0$ fulfilling $2k\le \kappa$, then $\textbf{p}_{2k+1}$ and $\textbf{r}_{2k+1}$ are ${q\times q}$ matrix polynomials with degree $k+1$ and leading coefficient matrix $I_{q}$, satisfying $\textbf{p}_{2k+1}\left( {\alpha }\right) =O_{{q\times q}}$ and $\textbf{r}_{2k+1}\left( {\alpha }\right) =O_{{q\times q}}$. Consequently, for each $k\in \mathbb {N}_0$ fulfilling $2k\le \kappa$, there exist unique ${q\times q}$ matrix polynomials $\textbf{b}_{{{\mathord {\circ }},k}}$ and $\textbf{d}_{{{\mathord {\circ }},k}}$ with degree k and leading coefficient matrix $I_{q}$, satisfying

$$\begin{aligned} \textbf{p}_{2k+1}\left( {z}\right) =\left( {z-\alpha }\right) \textbf{b}_{{{\mathord {\circ }},k}}\left( {z}\right) \qquad \text {and}\qquad \textbf{r}_{2k+1}\left( {z}\right) =\left( {z-\alpha }\right) \textbf{d}_{{{\mathord {\circ }},k}}\left( {z}\right) \qquad \text {for all }z \in \mathbb {C}. \end{aligned}$$

(12.1)

Taking into account Notation 10.3, it is readily checked that, for each $k\in \mathbb {N}_0$ fulfilling $2k\le \kappa$, then

$$\begin{aligned} \textbf{a}_{{{\mathord {\circ }},k}}:=\textbf{b}_{{{\mathord {\circ }},k}}^{\llbracket a\rrbracket }\quad \text {and}\quad \textbf{c}_{{{\mathord {\circ }},k}}:=\textbf{d}_{{{\mathord {\circ }},k}}^{\langle a\rangle } \end{aligned}$$

(12.2)

are ${q\times q}$ matrix polynomials of degree not greater than $k-1$, where the sequence is given via (7.1). The quadruple $\left[ {(\textbf{a}_{{{\mathord {\circ }},k}})_{k=0}^{\ddot{\kappa }},(\textbf{b}_{{{\mathord {\circ }},k}})_{k=0}^{\ddot{\kappa }},(\textbf{c}_{{{\mathord {\circ }},k}})_{k=0}^{\ddot{\kappa }},(\textbf{d}_{{{\mathord {\circ }},k}})_{k=0}^{\ddot{\kappa }}} \right]$ will be called the second ${[\alpha ,\infty )}$-quadruple of matrix polynomials (short: second ${[\alpha ,\infty )}$-QMP) associated with $(s_j)_{j=0}^{\kappa }$.

Remark 12.2

Suppose $\kappa \ge 1$. Let $(s_j)_{j=0}^{\kappa }$ be a sequence of Hermitian complex ${q\times q}$ matrices and let $k\in \mathbb {N}_0$ be such that $2k\le \kappa$. In view of (12.1) and Lemma 10.10, for all $z\in \mathbb {C}$, then $\left( {z-\alpha }\right) \left[ {\textbf{b}_{{{\mathord {\circ }},k}}\left( {\overline{z}}\right) } \right] ^*=\left[ {\left( {\overline{z}-\alpha }\right) \textbf{b}_{{{\mathord {\circ }},k}}\left( {\overline{z}}\right) } \right] ^*=\left[ {\textbf{p}_{2k+1}\left( {\overline{z}}\right) } \right] ^*=\textbf{r}_{2k+1}\left( {z}\right) =\left( {z-\alpha }\right) \textbf{d}_{{{\mathord {\circ }},k}}\left( {z}\right)$, implying $\left[ {\textbf{b}_{{{\mathord {\circ }},k}}\left( {\overline{z}}\right) } \right] ^*=\textbf{d}_{{{\mathord {\circ }},k}}\left( {z}\right)$. According to (2.2), hence $\textbf{b}_{{{\mathord {\circ }},k}}^{\mathord {\vee }}=\textbf{d}_{{{\mathord {\circ }},k}}$. Regarding (7.1), furthermore is a sequence of Hermitian matrices. Hence, using (12.2) and Lemma 10.4, we can conclude $\textbf{a}_{{{\mathord {\circ }},k}}^{\mathord {\vee }}=\left( {\textbf{b}_{{{\mathord {\circ }},k}}^{\llbracket a\rrbracket }}\right) ^{\mathord {\vee }}=\left( {\textbf{b}_{{{\mathord {\circ }},k}}^{\mathord {\vee }}}\right) ^{\langle a\rangle }=\textbf{d}_{{{\mathord {\circ }},k}}^{\langle a\rangle }=\textbf{c}_{{{\mathord {\circ }},k}}$. According to (2.2), thus $\left[ {\textbf{a}_{{{\mathord {\circ }},k}}\left( {\overline{z}}\right) } \right] ^*=\textbf{c}_{{{\mathord {\circ }},k}}\left( {z}\right)$ for all $z\in \mathbb {C}$.

Lemma 12.3

Suppose $\kappa \ge 1$. Let $(s_j)_{j=0}^{\kappa }\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,\kappa ,\alpha }$ and let $k \in \mathbb {N}_0$ be such that $2 k \le \kappa$. Then $\textbf{q}_{2k+1}=\textbf{a}_{{{\mathord {\circ }},k}}+s_{0}\textbf{b}_{{{\mathord {\circ }},k}}$ and $\textbf{t}_{2k+1}=\textbf{c}_{{{\mathord {\circ }},k}}+\textbf{d}_{{{\mathord {\circ }},k}}s_{0}$.

Proof

Remark 12.1 shows $\deg \textbf{b}_{{{\mathord {\circ }},k}}=k$. Regarding (12.1), thus the application of Lemma 10.5 to $\textbf{b}_{{{\mathord {\circ }},k}}$ yields $\textbf{p}_{2k+1}^{\llbracket s\rrbracket }=\textbf{b}_{{{\mathord {\circ }},k}}^{\llbracket a\rrbracket }+s_{0}\textbf{b}_{{{\mathord {\circ }},k}}$ if $k\ge 1$ and $\textbf{p}_{2k+1}^{\llbracket s\rrbracket }=s_{0}\textbf{b}_{{{\mathord {\circ }},k}}$ if $k=0$. In the case $k=0$, according to Notation 10.3, we have $\textbf{b}_{{{\mathord {\circ }},k}}^{\llbracket a\rrbracket }\left( {z}\right) =O_{{q\times q}}$ for all $z\in \mathbb {C}$. Using Proposition 10.13 and (12.2), consequently $\textbf{q}_{2k+1}=\textbf{a}_{{{\mathord {\circ }},k}}+s_{0}\textbf{b}_{{{\mathord {\circ }},k}}$ follows. Remark 9.3 shows that $(s_j)_{j=0}^{\kappa }$ is a sequence of Hermitian matrices. Taking additionally into account Lemma 10.10 and Remark 12.2, we conclude $\textbf{t}_{2k+1}\left( {z}\right) =\left[ {\textbf{q}_{2k+1}\left( {\overline{z}}\right) } \right] ^*=\left[ {\textbf{a}_{{{\mathord {\circ }},k}}\left( {\overline{z}}\right) } \right] ^*+\left[ {\textbf{b}_{{{\mathord {\circ }},k}}\left( {\overline{z}}\right) } \right] ^*s_{0}^*=\textbf{c}_{{{\mathord {\circ }},k}}\left( {z}\right) +\textbf{d}_{{{\mathord {\circ }},k}}\left( {z}\right) s_{0}$ for all $z\in \mathbb {C}$. $\square$

Remark 12.4

Suppose $\kappa \ge 1$. Let $(s_j)_{j=0}^{\kappa }\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,\kappa ,\alpha }$ and let $k\in \mathbb {N}_0$ be such that $2k\le \kappa$. In view of (12.1) and Remark 10.12, then $\mathcal {Z}\left( {\det \textbf{d}_{{{\mathord {\circ }},k}}}\right) =\mathcal {Z}\left( {\det \textbf{b}_{{{\mathord {\circ }},k}}}\right) \subseteq {[\alpha ,\infty )}$.

Lemma 12.5

(cf. [13, Lem. 14.6]) Suppose $\kappa \ge 1$. Let $(s_j)_{j=0}^{\kappa }$ be a sequence of complex ${q\times q}$ matrices and let $z\in \mathbb {C}$. Then $\textbf{b}_{0}\left( {z}\right) =I_{q}$ and $\textbf{b}_{{{\mathord {\circ }},0}}\left( {z}\right) =I_{q}$. Furthermore, $\textbf{b}_{k}\left( {z}\right) =\left( {z-\alpha }\right) \textbf{b}_{{{\mathord {\circ }},k-1}}\left( {z}\right) -\textbf{b}_{k-1}\left( {z}\right) \mathfrak {k}_{2k-2}^{\mathord {+}}\mathfrak {k}_{2k-1}$ for each $k \in \mathbb {N}$ fulfilling $2 k -1\le \kappa$ and, in the case $\kappa \ge 2$, moreover $\textbf{b}_{{{\mathord {\circ }},k}}\left( {z}\right) =\textbf{b}_{k}\left( {z}\right) -\textbf{b}_{{{\mathord {\circ }},k-1}}\left( {z}\right) \mathfrak {k}_{2k-1}^{\mathord {+}}\mathfrak {k}_{2 k }$ for each $k \in \mathbb {N}$ fulfilling $2 k \le \kappa$.

Remark 12.6

Suppose $\kappa \ge 2$. Let $(s_j)_{j=0}^{\kappa }$ be a sequence of complex ${q\times q}$ matrices and let $k\in \mathbb {N}$ be such that $2k\le \kappa$. According to Lemma 12.5, then $\textbf{b}_{k}=\textbf{b}_{{{\mathord {\circ }},k}}+\textbf{b}_{{{\mathord {\circ }},k-1}}\mathfrak {k}_{2k-1}^{\mathord {+}}\mathfrak {k}_{2k}$.

Remark 12.7

Suppose $\kappa \ge 2$. Let $(s_j)_{j=0}^{\kappa }\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,\kappa ,\alpha }$ and let $k \in \mathbb {N}$ be such that $2 k \le \kappa$. Then

$$\begin{aligned} \left( {\epsilon -\alpha }\right) \textbf{a}_{k}-s_{0}\textbf{b}_{k} =\textbf{a}_{{{\mathord {\circ }},k}}+\textbf{a}_{{{\mathord {\circ }},k-1}}\mathfrak {k}_{2k-1}^{\mathord {+}}\mathfrak {k}_{2 k }. \end{aligned}$$

Indeed, using Lemma 12.3, (10.4), Notations 10.6 and 5.5, Lemma 11.3, and Remark 12.6, we can infer

$$\begin{aligned}\begin{aligned} \textbf{a}_{{{\mathord {\circ }},k}}&=\textbf{q}_{2k+1}-s_{0}\textbf{b}_{{{\mathord {\circ }},k}} =\varepsilon _{2 k }\textbf{q}_{2 k }-\textbf{q}_{2k-1}\mathfrak {k}_{2k-1}^{\mathord {+}}\mathfrak {k}_{2 k }-s_{0}\textbf{b}_{{{\mathord {\circ }},k}}\\&=\left( {\epsilon -\alpha }\right) \textbf{a}_{k}-\left( {\textbf{a}_{{{\mathord {\circ }},k-1}}+s_{0}\textbf{b}_{{{\mathord {\circ }},k-1}}}\right) \mathfrak {k}_{2k-1}^{\mathord {+}}\mathfrak {k}_{2 k }-s_{0}\textbf{b}_{{{\mathord {\circ }},k}}\\&=\left( {\epsilon -\alpha }\right) \textbf{a}_{k}-\textbf{a}_{{{\mathord {\circ }},k-1}}\mathfrak {k}_{2k-1}^{\mathord {+}}\mathfrak {k}_{2 k }-s_{0}\left( {\textbf{b}_{{{\mathord {\circ }},k-1}}\mathfrak {k}_{2k-1}^{\mathord {+}}\mathfrak {k}_{2 k }+\textbf{b}_{{{\mathord {\circ }},k}}}\right) \\&=\left( {\epsilon -\alpha }\right) \textbf{a}_{k}-\textbf{a}_{{{\mathord {\circ }},k-1}}\mathfrak {k}_{2k-1}^{\mathord {+}}\mathfrak {k}_{2 k }-s_{0}\textbf{b}_{k}. \end{aligned} \end{aligned}$$

Lemma 12.8

Suppose $\kappa \ge 1$. Let $(s_j)_{j=0}^{\kappa }\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,\kappa ,\alpha }$. For all $z\in \mathbb {C}$, then $\textbf{b}_{{{\mathord {\circ }},0}}\left( {z}\right) =I_{q}$ and $\textbf{d}_{{{\mathord {\circ }},0}}\left( {z}\right) =I_{q}$. If $\kappa \ge 2$, then $\textbf{b}_{{{\mathord {\circ }},1}}=\left( {\epsilon -\alpha }\right) I_{q}-\left( {\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{1}+\mathfrak {k}_{1}^{\mathord {+}}\mathfrak {k}_{2}}\right)$ and $\textbf{d}_{{{\mathord {\circ }},1}}=\left( {\epsilon -\alpha }\right) I_{q}-\left( {\mathfrak {k}_{1}\mathfrak {k}_{0}^{\mathord {+}}+\mathfrak {k}_{2}\mathfrak {k}_{1}^{\mathord {+}}}\right)$. If $\kappa \ge 4$, for all $k\in \mathbb {Z}_{2,\infty }$ fulfilling $2k\le \kappa$, furthermore

$$\begin{aligned} \textbf{b}_{{{\mathord {\circ }},k}} =\textbf{b}_{{{\mathord {\circ }},k-1}}\left[ {\left( {\epsilon -\alpha }\right) I_{q}-\left( {\mathfrak {k}_{2k-2}^{\mathord {+}}\mathfrak {k}_{2k-1}+\mathfrak {k}_{2k-1}^{\mathord {+}}\mathfrak {k}_{2k}}\right) } \right] -\textbf{b}_{{{\mathord {\circ }},k-2}}\mathfrak {k}_{2k-3}^{\mathord {+}}\mathfrak {k}_{2k-1}\nonumber \\ \end{aligned}$$

(12.3)

and

$$\begin{aligned} \textbf{d}_{{{\mathord {\circ }},k}} =\left[ {\left( {\epsilon -\alpha }\right) I_{q}-\left( {\mathfrak {k}_{2k-1}\mathfrak {k}_{2k-2}^{\mathord {+}}+\mathfrak {k}_{2k}\mathfrak {k}_{2k-1}^{\mathord {+}}}\right) } \right] \textbf{d}_{{{\mathord {\circ }},k-1}}-\mathfrak {k}_{2k-1}\mathfrak {k}_{2k-3}^{\mathord {+}}\textbf{d}_{{{\mathord {\circ }},k-2}}.\nonumber \\ \end{aligned}$$

(12.4)

Proof

In view of (12.1) and (10.3), we have $\textbf{b}_{{{\mathord {\circ }},0}}\left( {z}\right) =I_{q}$ and $\textbf{d}_{{{\mathord {\circ }},0}}\left( {z}\right) =I_{q}$ for all $z\in \mathbb {C}$. If $\kappa \ge 2$, we can use (12.1), Lemma 10.15, and (10.3) to obtain $\textbf{b}_{{{\mathord {\circ }},1}}=\left( {\epsilon -\alpha }\right) I_{q}-\left( {\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{1}+\mathfrak {k}_{1}^{\mathord {+}}\mathfrak {k}_{2}}\right)$ and $\textbf{d}_{{{\mathord {\circ }},1}}=\left( {\epsilon -\alpha }\right) I_{q}-\left( {\mathfrak {k}_{1}\mathfrak {k}_{0}^{\mathord {+}}+\mathfrak {k}_{2}\mathfrak {k}_{1}^{\mathord {+}}}\right)$. Now suppose $\kappa \ge 4$. Let $k\in \mathbb {Z}_{2,\infty }$ be such that $2k\le \kappa$. Remark 9.3 yields then $\mathfrak {k}_{2k-2}\mathfrak {k}_{2k-2}^{\mathord {+}}\mathfrak {k}_{2k-1}=\mathfrak {k}_{2k-1}$ and $\mathfrak {k}_{2k-1}\mathfrak {k}_{2k-2}^{\mathord {+}}\mathfrak {k}_{2k-2}=\mathfrak {k}_{2k-1}$. Taking additionally into account (12.1) and Lemma 10.15, we can conclude then (12.3) and (12.4). $\square$

Lemma 12.9

Suppose $\kappa \ge 1$. Let $(s_j)_{j=0}^{\kappa }\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,\kappa ,\alpha }$. For all $z\in \mathbb {C}$, then $\textbf{a}_{{{\mathord {\circ }},0}}\left( {z}\right) =O_{{q\times q}}$ and $\textbf{c}_{{{\mathord {\circ }},0}}\left( {z}\right) =O_{{q\times q}}$. If $\kappa \ge 2$, then $\textbf{a}_{{{\mathord {\circ }},1}}\left( {z}\right) =\mathfrak {k}_{1}$ and $\textbf{c}_{{{\mathord {\circ }},1}}\left( {z}\right) =\mathfrak {k}_{1}$ for all $z\in \mathbb {C}$. If $\kappa \ge 4$, for all $k\in \mathbb {Z}_{2,\infty }$ fulfilling $2k\le \kappa$, furthermore

$$\begin{aligned} \textbf{a}_{{{\mathord {\circ }},k}}&=\textbf{a}_{{{\mathord {\circ }},k-1}}\left[ {\left( {\epsilon -\alpha }\right) I_{q}-\left( {\mathfrak {k}_{2k-2}^{\mathord {+}}\mathfrak {k}_{2k-1}+\mathfrak {k}_{2k-1}^{\mathord {+}}\mathfrak {k}_{2k}}\right) } \right] -\textbf{a}_{{{\mathord {\circ }},k-2}}\mathfrak {k}_{2k-3}^{\mathord {+}}\mathfrak {k}_{2k-1} \end{aligned}$$

and

$$\begin{aligned} \textbf{c}_{{{\mathord {\circ }},k}}&=\left[ {\left( {\epsilon -\alpha }\right) I_{q}-\left( {\mathfrak {k}_{2k-1}\mathfrak {k}_{2k-2}^{\mathord {+}}+\mathfrak {k}_{2k}\mathfrak {k}_{2k-1}^{\mathord {+}}}\right) } \right] \textbf{c}_{{{\mathord {\circ }},k-1}}-\mathfrak {k}_{2k-1}\mathfrak {k}_{2k-3}^{\mathord {+}}\textbf{c}_{{{\mathord {\circ }},k-2}}. \end{aligned}$$

Proof

Lemma 12.8 shows $\textbf{b}_{{{\mathord {\circ }},0}}\left( {z}\right) =I_{q}$ and $\textbf{d}_{{{\mathord {\circ }},0}}\left( {z}\right) =I_{q}$ for all $z\in \mathbb {C}$. According to (10.3), we have $\textbf{q}_{1}\left( {z}\right) =\mathfrak {k}_{0}$ and $\textbf{t}_{1}\left( {z}\right) =\mathfrak {k}_{0}$ for all $z\in \mathbb {C}$. Taking additionally into account Lemma 12.3 and $\mathfrak {k}_{0}=s_{0}$, we can conclude $\textbf{a}_{{{\mathord {\circ }},0}}\left( {z}\right) =\textbf{q}_{1}\left( {z}\right) -s_{0}\textbf{b}_{{{\mathord {\circ }},0}}\left( {z}\right) =\mathfrak {k}_{0}-s_{0}\cdot I_{q}=O_{{q\times q}}$ and $\textbf{c}_{{{\mathord {\circ }},0}}\left( {z}\right) =\textbf{t}_{1}\left( {z}\right) -\textbf{d}_{{{\mathord {\circ }},0}}\left( {z}\right) s_{0}=\mathfrak {k}_{0}-I_{q}\cdot s_{0}=O_{{q\times q}}$ for all $z\in \mathbb {C}$. Now suppose $\kappa \ge 2$. Using Lemma 10.16 and regarding (10.3) and Notation 5.5, we can infer $\textbf{q}_{3}\left( {z}\right) =\mathfrak {k}_{0}\left[ {\left( {z-\alpha }\right) I_{q}-\mathfrak {k}_{1}^{\mathord {+}}\mathfrak {k}_{2}} \right]$ and $\textbf{t}_{3}\left( {z}\right) =\left[ {\left( {z-\alpha }\right) I_{q}-\mathfrak {k}_{2}\mathfrak {k}_{1}^{\mathord {+}}} \right] \mathfrak {k}_{0}$ for all $z\in \mathbb {C}$. Lemma 12.8 shows $\textbf{b}_{{{\mathord {\circ }},1}}\left( {z}\right) =\left( {z-\alpha }\right) I_{q}-\left( {\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{1}+\mathfrak {k}_{1}^{\mathord {+}}\mathfrak {k}_{2}}\right)$ and $\textbf{d}_{{{\mathord {\circ }},1}}\left( {z}\right) =\left( {z-\alpha }\right) I_{q}-\left( {\mathfrak {k}_{1}\mathfrak {k}_{0}^{\mathord {+}}+\mathfrak {k}_{2}\mathfrak {k}_{1}^{\mathord {+}}}\right)$ for all $z\in \mathbb {C}$. Taking additionally into account Lemma 12.3, $\mathfrak {k}_{0}=s_{0}$, and that Remark 9.3 yields $\mathfrak {k}_{0}\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{1}=\mathfrak {k}_{1}$ and $\mathfrak {k}_{1}\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}=\mathfrak {k}_{1}$, for all $z\in \mathbb {C}$, we can conclude

$$\begin{aligned}\begin{aligned} \textbf{a}_{{{\mathord {\circ }},1}}\left( {z}\right)&=\textbf{q}_{3}\left( {z}\right) -s_{0}\textbf{b}_{{{\mathord {\circ }},1}}\left( {z}\right) \\&=\mathfrak {k}_{0}\left[ {\left( {z-\alpha }\right) I_{q}-\mathfrak {k}_{1}^{\mathord {+}}\mathfrak {k}_{2}} \right] -\mathfrak {k}_{0}\left[ {\left( {z-\alpha }\right) I_{q}-\left( {\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{1}+\mathfrak {k}_{1}^{\mathord {+}}\mathfrak {k}_{2}}\right) } \right] =\mathfrak {k}_{1} \end{aligned}\end{aligned}$$

and

$$\begin{aligned}\begin{aligned} \textbf{c}_{{{\mathord {\circ }},1}}\left( {z}\right)&=\textbf{t}_{3}\left( {z}\right) -\textbf{d}_{{{\mathord {\circ }},1}}\left( {z}\right) s_{0}\\&=\left[ {\left( {z-\alpha }\right) I_{q}-\mathfrak {k}_{2}\mathfrak {k}_{1}^{\mathord {+}}} \right] \mathfrak {k}_{0}-\left[ {\left( {z-\alpha }\right) I_{q}-\left( {\mathfrak {k}_{1}\mathfrak {k}_{0}^{\mathord {+}}+\mathfrak {k}_{2}\mathfrak {k}_{1}^{\mathord {+}}}\right) } \right] \mathfrak {k}_{0} =\mathfrak {k}_{1}. \end{aligned}\end{aligned}$$

Now suppose $\kappa \ge 4$. We consider an arbitrary $k\in \mathbb {Z}_{2,\infty }$ fulfilling $2k\le \kappa$. Remark 9.3 yields $\mathfrak {k}_{2k-2}\mathfrak {k}_{2k-2}^{\mathord {+}}\mathfrak {k}_{2k-1}=\mathfrak {k}_{2k-1}$ and $\mathfrak {k}_{2k-1}\mathfrak {k}_{2k-2}^{\mathord {+}}\mathfrak {k}_{2k-2}=\mathfrak {k}_{2k-1}$. From Lemma 10.16 we get then

$$\begin{aligned} \textbf{q}_{2k+1} =\textbf{q}_{2k-1}\left[ {\left( {\epsilon -\alpha }\right) I_{q}-\left( {\mathfrak {k}_{2k-2}^{\mathord {+}}\mathfrak {k}_{2k-1}+\mathfrak {k}_{2k-1}^{\mathord {+}}\mathfrak {k}_{2k}}\right) } \right] -\textbf{q}_{2k-3}\mathfrak {k}_{2k-3}^{\mathord {+}}\mathfrak {k}_{2k-1}\nonumber \\ \end{aligned}$$

(12.5)

and

$$\begin{aligned} \textbf{t}_{2k+1} =\left[ {\left( {\epsilon - \alpha }\right) I_{q}-\left( {\mathfrak {k}_{2k-1}\mathfrak {k}_{2k-2}^{\mathord {+}}+\mathfrak {k}_{2k}\mathfrak {k}_{2k-1}^{\mathord {+}}}\right) } \right] \textbf{t}_{2k-1}-\mathfrak {k}_{2k-1}\mathfrak {k}_{2k-3}^{\mathord {+}}\textbf{t}_{2k-3}.\nonumber \\ \end{aligned}$$

(12.6)

Lemma 12.8 yields (12.3) and (12.4). By virtue of Lemma 12.3, (12.5), and (12.3), we obtain

$$\begin{aligned}\begin{aligned}&\textbf{a}_{{{\mathord {\circ }},k}} =\textbf{q}_{2k+1}-s_{0}\textbf{b}_{{{\mathord {\circ }},k}}\\&=\textbf{q}_{2k-1}\left[ {\left( {\epsilon -\alpha }\right) I_{q}-\left( {\mathfrak {k}_{2k-2}^{\mathord {+}}\mathfrak {k}_{2k-1}+\mathfrak {k}_{2k-1}^{\mathord {+}}\mathfrak {k}_{2k}}\right) } \right] -\textbf{q}_{2k-3}\mathfrak {k}_{2k-3}^{\mathord {+}}\mathfrak {k}_{2k-1}\\&\qquad -s_{0}\left( {\textbf{b}_{{{\mathord {\circ }},k-1}}\left[ {\left( {\epsilon -\alpha }\right) I_{q}-\left( {\mathfrak {k}_{2k-2}^{\mathord {+}}\mathfrak {k}_{2k-1}+\mathfrak {k}_{2k-1}^{\mathord {+}}\mathfrak {k}_{2k}}\right) } \right] -\textbf{b}_{{{\mathord {\circ }},k-2}}\mathfrak {k}_{2k-3}^{\mathord {+}}\mathfrak {k}_{2k-1}}\right) \\&=\left( {\textbf{q}_{2k-1}-s_{0}\textbf{b}_{{{\mathord {\circ }},k-1}}}\right) \left[ {\left( {\epsilon -\alpha }\right) I_{q}-\left( {\mathfrak {k}_{2k-2}^{\mathord {+}}\mathfrak {k}_{2k-1}+\mathfrak {k}_{2k-1}^{\mathord {+}}\mathfrak {k}_{2k}}\right) } \right] \\&\qquad -\left( {\textbf{q}_{2k-3}-s_{0}\textbf{b}_{{{\mathord {\circ }},k-2}}}\right) \mathfrak {k}_{2k-3}^{\mathord {+}}\mathfrak {k}_{2k-1}\\&=\textbf{a}_{{{\mathord {\circ }},k-1}}\left[ {\left( {\epsilon -\alpha }\right) I_{q}-\left( {\mathfrak {k}_{2k-2}^{\mathord {+}}\mathfrak {k}_{2k-1}+\mathfrak {k}_{2k-1}^{\mathord {+}}\mathfrak {k}_{2k}}\right) } \right] -\textbf{a}_{{{\mathord {\circ }},k-2}}\mathfrak {k}_{2k-3}^{\mathord {+}}\mathfrak {k}_{2k-1}. \end{aligned}\end{aligned}$$

Using Lemma 12.3, (12.6), and (12.4), we get analogously the equation $\textbf{c}_{{{\mathord {\circ }},k}}=\left[ {\left( {\epsilon -\alpha }\right) I_{q}-\left( {\mathfrak {k}_{2k-1}\mathfrak {k}_{2k-2}^{\mathord {+}}+\mathfrak {k}_{2k}\mathfrak {k}_{2k-1}^{\mathord {+}}}\right) } \right] \textbf{c}_{{{\mathord {\circ }},k-1}}-\mathfrak {k}_{2k-1}\mathfrak {k}_{2k-3}^{\mathord {+}}\textbf{c}_{{{\mathord {\circ }},k-2}}$. $\square$

Remark 12.10

Suppose $\kappa \ge 4$. Let $(s_j)_{j=0}^{\kappa }\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,\kappa ,\alpha }$ and let $k\in \mathbb {N}$ be such that $2k+2\le \kappa$. In view of Lemmata 12.9 and 12.8, then

and

Lemma 12.11

Suppose $\kappa \ge 2$. Let $(s_j)_{j=0}^{\kappa }\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,\kappa ,\alpha }$. For each $k\in \mathbb {N}_0$ fulfilling $2k+2\le \kappa$ and all $z\in \mathbb {C}\backslash {[\alpha ,\infty )}$, then

$$\begin{aligned} \det \textbf{b}_{{{\mathord {\circ }},k}}\left( {z}\right) \ne 0,\; \det \textbf{d}_{{{\mathord {\circ }},k}}\left( {z}\right) \ne 0,\; \det \textbf{b}_{{{\mathord {\circ }},k+1}}\left( {z}\right) \ne 0, \text { and } \det \textbf{d}_{{{\mathord {\circ }},k+1}}\left( {z}\right) \ne 0 \nonumber \\ \end{aligned}$$

(12.7)

as well as

$$\begin{aligned} \mathfrak {h}_{{\alpha ,2k}}\left[ {\textbf{b}_{{{\mathord {\circ }},k}}\left( {z}\right) } \right] ^{-1}\textbf{b}_{{{\mathord {\circ }},k+1}}\left( {z}\right)&=\textbf{d}_{{{\mathord {\circ }},k+1}}\left( {z}\right) \left[ {\textbf{d}_{{{\mathord {\circ }},k}}\left( {z}\right) } \right] ^{-1}\mathfrak {h}_{{\alpha ,2k}} \end{aligned}$$

(12.8)

and

$$\begin{aligned} \mathfrak {h}_{{\alpha ,2k}}\left[ {\textbf{b}_{{{\mathord {\circ }},k+1}}\left( {z}\right) } \right] ^{-1}\textbf{b}_{{{\mathord {\circ }},k}}\left( {z}\right)&=\textbf{d}_{{{\mathord {\circ }},k}}\left( {z}\right) \left[ {\textbf{d}_{{{\mathord {\circ }},k+1}}\left( {z}\right) } \right] ^{-1}\mathfrak {h}_{{\alpha ,2k}}. \end{aligned}$$

(12.9)

Proof

Throughout this proof, we consider an arbitrary $z\in \mathbb {C}\backslash {[\alpha ,\infty )}$. From Remark 12.4 we know that (12.7) is valid for each $k\in \mathbb {N}_0$ fulfilling $2k+2\le \kappa$. We now proceed by mathematical induction. Using Lemma 12.8, Notation 5.5, and Lemma 9.7, we obtain

$$\begin{aligned}\begin{aligned} \mathfrak {h}_{{\alpha ,0}}\left[ {\textbf{b}_{{{\mathord {\circ }},0}}\left( {z}\right) } \right] ^{-1}\textbf{b}_{{{\mathord {\circ }},1}}\left( {z}\right)&=\mathfrak {h}_{{\alpha ,0}}\left[ {\left( {z-\alpha }\right) I_{q}-\left( {\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{1}+\mathfrak {k}_{1}^{\mathord {+}}\mathfrak {k}_{2}}\right) } \right] \\&=z\mathfrak {h}_{{\alpha ,0}}-\mathfrak {h}_{{\alpha ,0}}\left( {\alpha I_{q}+\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{1}+\mathfrak {k}_{1}^{\mathord {+}}\mathfrak {k}_{2}}\right) =z\mathfrak {h}_{{\alpha ,0}}-\mathfrak {h}_{{\alpha ,1}} \end{aligned} \end{aligned}$$

and, analogously, $\textbf{d}_{{{\mathord {\circ }},1}}\left( {z}\right) \left[ {\textbf{d}_{{{\mathord {\circ }},0}}\left( {z}\right) } \right] ^{-1}\mathfrak {h}_{{\alpha ,0}}=z\mathfrak {h}_{{\alpha ,0}}-\mathfrak {h}_{{\alpha ,1}}$. Therefore, (12.8) follows for $k=0$, which in turn implies (12.9) for $k=0$. Now suppose $\kappa \ge 4$. We already know that there exists an integer $\ell \in \mathbb {N}$ fulfilling $2\ell +2\le \kappa$ such that (12.8) and (12.9) are valid for $k=\ell -1$. Using Lemma 12.8, Notation 5.5, and Lemma 9.7, we obtain

$$\begin{aligned}\begin{aligned}&\mathfrak {h}_{{\alpha ,2\ell }}\left[ {\textbf{b}_{{{\mathord {\circ }},\ell }}\left( {z}\right) } \right] ^{-1}\textbf{b}_{{{\mathord {\circ }},\ell +1}}\left( {z}\right) \\&=\mathfrak {h}_{{\alpha ,2\ell }}\left[ {\textbf{b}_{{{\mathord {\circ }},\ell }}\left( {z}\right) } \right] ^{-1}\\&\qquad \times \left( {\textbf{b}_{{{\mathord {\circ }},\ell }}\left( {z}\right) \left[ {\left( {z-\alpha }\right) I_{q}-\left( {\mathfrak {k}_{2\ell }^{\mathord {+}}\mathfrak {k}_{2\ell +1}+\mathfrak {k}_{2\ell +1}^{\mathord {+}}\mathfrak {k}_{2\ell +2}}\right) } \right] -\textbf{b}_{{{\mathord {\circ }},\ell -1}}\left( {z}\right) \mathfrak {k}_{2\ell -1}^{\mathord {+}}\mathfrak {k}_{2\ell +1}}\right) \\&=z\mathfrak {h}_{{\alpha ,2\ell }}-\mathfrak {h}_{{\alpha ,2\ell }}\left( {\alpha I_{q}+\mathfrak {k}_{2\ell }^{\mathord {+}}\mathfrak {k}_{2\ell +1}+\mathfrak {k}_{2\ell +1}^{\mathord {+}}\mathfrak {k}_{2\ell +2}}\right) \\&\qquad -\mathfrak {h}_{{\alpha ,2\ell }}\left[ {\textbf{b}_{{{\mathord {\circ }},\ell }}\left( {z}\right) } \right] ^{-1}\textbf{b}_{{{\mathord {\circ }},\ell -1}}\left( {z}\right) \mathfrak {k}_{2\ell -1}^{\mathord {+}}\mathfrak {k}_{2\ell +1}\\&=z\mathfrak {h}_{{\alpha ,2\ell }}-\mathfrak {h}_{{\alpha ,2\ell +1}}-\mathfrak {h}_{{\alpha ,2\ell }}\left[ {\textbf{b}_{{{\mathord {\circ }},\ell }}\left( {z}\right) } \right] ^{-1}\textbf{b}_{{{\mathord {\circ }},\ell -1}}\left( {z}\right) \mathfrak {k}_{2\ell -1}^{\mathord {+}}\mathfrak {k}_{2\ell +1} \end{aligned}\end{aligned}$$

and, analogously, $\textbf{d}_{{{\mathord {\circ }},\ell +1}}\left( {z}\right) \left[ {\textbf{d}_{{{\mathord {\circ }},\ell }}\left( {z}\right) } \right] ^{-1}\mathfrak {h}_{{\alpha ,2\ell }}=z\mathfrak {h}_{{\alpha ,2\ell }}-\mathfrak {h}_{{\alpha ,2\ell +1}}-\mathfrak {k}_{2\ell +1}\mathfrak {k}_{2\ell -1}^{\mathord {+}}\textbf{d}_{{{\mathord {\circ }},\ell -1}}\left( {z}\right) \left[ {\textbf{d}_{{{\mathord {\circ }},\ell }}\left( {z}\right) } \right] ^{-1}\mathfrak {h}_{{\alpha ,2\ell }}$. In view of Remark 7.6, we have . Thus, Remark 5.3 yields $\mathfrak {h}_{{\alpha ,2\ell -2}}\mathfrak {h}_{{\alpha ,2\ell -2}}^{\mathord {+}}\mathfrak {h}_{{\alpha ,2\ell }}=\mathfrak {h}_{{\alpha ,2\ell }}$ and $\mathfrak {h}_{{\alpha ,2\ell }}\mathfrak {h}_{{\alpha ,2\ell -2}}^{\mathord {+}}\mathfrak {h}_{{\alpha ,2\ell -2}}=\mathfrak {h}_{{\alpha ,2\ell }}$. Furthermore, Remark 9.2 provides $\mathfrak {k}_{2\ell -1}=\mathfrak {h}_{{\alpha ,2\ell -2}}$ and $\mathfrak {k}_{2\ell +1}=\mathfrak {h}_{{\alpha ,2\ell }}$. Taking additionally into account (12.9) for $k=\ell -1$, we can conclude

$$\begin{aligned}\begin{aligned}&\mathfrak {h}_{{\alpha ,2\ell }}\left[ {\textbf{b}_{{{\mathord {\circ }},\ell }}\left( {z}\right) } \right] ^{-1}\textbf{b}_{{{\mathord {\circ }},\ell -1}}\left( {z}\right) \mathfrak {k}_{2\ell -1}^{\mathord {+}}\mathfrak {k}_{2\ell +1}\\&=\mathfrak {h}_{{\alpha ,2\ell }}\mathfrak {h}_{{\alpha ,2\ell -2}}^{\mathord {+}}\mathfrak {h}_{{\alpha ,2\ell -2}}\left[ {\textbf{b}_{{{\mathord {\circ }},\ell }}\left( {z}\right) } \right] ^{-1}\textbf{b}_{{{\mathord {\circ }},\ell -1}}\left( {z}\right) \mathfrak {h}_{{\alpha ,2\ell -2}}^{\mathord {+}}\mathfrak {h}_{{\alpha ,2\ell }}\\&=\mathfrak {h}_{{\alpha ,2\ell }}\mathfrak {h}_{{\alpha ,2\ell -2}}^{\mathord {+}}\textbf{d}_{{{\mathord {\circ }},\ell -1}}\left( {z}\right) \left[ {\textbf{d}_{{{\mathord {\circ }},\ell }}\left( {z}\right) } \right] ^{-1}\mathfrak {h}_{{\alpha ,2\ell -2}}\mathfrak {h}_{{\alpha ,2\ell -2}}^{\mathord {+}}\mathfrak {h}_{{\alpha ,2\ell }}\\&=\mathfrak {k}_{2\ell +1}\mathfrak {k}_{2\ell -1}^{\mathord {+}}\textbf{d}_{{{\mathord {\circ }},\ell -1}}\left( {z}\right) \left[ {\textbf{d}_{{{\mathord {\circ }},\ell }}\left( {z}\right) } \right] ^{-1}\mathfrak {h}_{{\alpha ,2\ell }}. \end{aligned}\end{aligned}$$

Therefore, (12.8) follows for $k=\ell$, which in turn implies (12.9) for $k=\ell$. Thus, (12.8) and (12.9) are proved for each $k\in \mathbb {N}_0$ fulfilling $2k+2\le \kappa$ by mathematical induction. $\square$

Lemma 12.12

Suppose $\kappa \ge 2$. Let $(s_j)_{j=0}^{\kappa }\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,\kappa ,\alpha }$, let $k\in \mathbb {N}_0$ be such that $2k+2\le \kappa$, and let $z\in \mathbb {C}\backslash {[\alpha ,\infty )}$. Then $\det \textbf{b}_{{{\mathord {\circ }},k}}\left( {z}\right) \ne 0$ and $\det \textbf{b}_{{{\mathord {\circ }},k+1}}\left( {z}\right) \ne 0$ as well as $\left( {\mathfrak {h}_{{\alpha ,2k}}\left[ {\textbf{b}_{{{\mathord {\circ }},k}}\left( {z}\right) } \right] ^{-1}\textbf{b}_{{{\mathord {\circ }},k+1}}\left( {z}\right) }\right) ^{\mathord {+}}=\mathfrak {h}_{{\alpha ,2k}}^{\mathord {+}}\mathfrak {h}_{{\alpha ,2k}}\left[ {\textbf{b}_{{{\mathord {\circ }},k+1}}\left( {z}\right) } \right] ^{-1}\textbf{b}_{{{\mathord {\circ }},k}}\left( {z}\right) \mathfrak {h}_{{\alpha ,2k}}^{\mathord {+}}$.

Proof

In view of Lemma 12.11, all the matrices $B_{k}:=\textbf{b}_{{{\mathord {\circ }},k}}\left( {z}\right)$, $D_{k}:=\textbf{d}_{{{\mathord {\circ }},k}}\left( {z}\right)$, $B_{ k +1}:=\textbf{b}_{{{\mathord {\circ }}, k +1}}\left( {z}\right)$, and $D_{ k +1}:=\textbf{d}_{{{\mathord {\circ }}, k +1}}\left( {z}\right)$ are invertible. Hence, the matrices $L:=D_{k}D_{ k +1}^{-1}$ and $R:=B_{ k +1}^{-1}B_{k}$ are invertible. Setting $M:=\mathfrak {h}_{{\alpha ,2 k }}$, $N:=LMR^{-1}$, and $X:=MR^{-1}$, we thus can apply Lemma A.15 to obtain $X^{\mathord {+}}=N^{\mathord {+}}NRM^{\mathord {+}}$. Since Lemma 12.11 provides $MR=LM$, we have $N=MRR^{-1}=M$ and, consequently, $X^{\mathord {+}}=M^{\mathord {+}}MB_{ k +1}^{-1}B_{k}M^{\mathord {+}}$. In view of $X=MR^{-1}=MB_{k}^{-1}B_{ k +1}$, the proof is complete.

Suppose $\kappa \ge 1$. Let $(s_j)_{j=0}^{\kappa }\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,\kappa ,\alpha }$. Because of Remark 7.6, then . In accordance with Notation 7.1, we denote in the sequel by $(\chi _{{\alpha ,j}})_{j=-1}^{\kappa -1}$ the sequence of $\chi$-functions given by Definition 6.1 for the sequence .

Lemma 12.13

Suppose $\kappa \ge 2$. Let $(s_j)_{j=0}^{\kappa }\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,\kappa ,\alpha }$. For each $k\in \mathbb {N}_0$ fulfilling $2k+2\le \kappa$ and all $z\in \mathbb {C}\backslash {[\alpha ,\infty )}$, then $\det \textbf{b}_{{{\mathord {\circ }},k}}\left( {z}\right) \ne 0$ and

$$\begin{aligned} \chi _{{\alpha ,2k+1}}\left( {z}\right) =\mathfrak {h}_{{\alpha ,2k}}\left[ {\textbf{b}_{{{\mathord {\circ }},k}}\left( {z}\right) } \right] ^{-1}\textbf{b}_{{{\mathord {\circ }},k+1}}\left( {z}\right) . \end{aligned}$$

(12.10)

Proof

Let $z\in \mathbb {C}\backslash {[\alpha ,\infty )}$. First observe that Remark 7.6 yields . Remark 12.4 shows $\det \textbf{b}_{{{\mathord {\circ }},n}}\left( {z}\right) \ne 0$ for each $n\in \mathbb {N}_0$ fulfilling $2n\le \kappa$. We now proceed by mathematical induction. As in the proof of Lemma 12.11 we can infer $\mathfrak {h}_{{\alpha ,0}}\left[ {\textbf{b}_{{{\mathord {\circ }},0}}\left( {z}\right) } \right] ^{-1}\textbf{b}_{{{\mathord {\circ }},1}}\left( {z}\right) =z\mathfrak {h}_{{\alpha ,0}}-\mathfrak {h}_{{\alpha ,1}}$, which in view of (6.1) implies (12.10) for $k=0$. Now suppose $\kappa \ge 4$ and that there exists an integer $\ell \in \mathbb {N}$ fulfilling $2\ell +2\le \kappa$ such that (12.10) is valid for $k=\ell -1$. As in the proof of Lemma 12.11 we can infer

$$\begin{aligned}{} & {} \mathfrak {h}_{{\alpha ,2\ell }}\left[ {\textbf{b}_{{{\mathord {\circ }},\ell }}\left( {z}\right) } \right] ^{-1}\textbf{b}_{{{\mathord {\circ }},\ell +1}}\left( {z}\right) \\{} & {} \quad =z\mathfrak {h}_{{\alpha ,2\ell }}-\mathfrak {h}_{{\alpha ,2\ell +1}}-\mathfrak {h}_{{\alpha ,2\ell }}\left[ {\textbf{b}_{{{\mathord {\circ }},\ell }}\left( {z}\right) } \right] ^{-1}\textbf{b}_{{{\mathord {\circ }},\ell -1}}\left( {z}\right) \mathfrak {k}_{2\ell -1}^{\mathord {+}}\mathfrak {k}_{2\ell +1}, \end{aligned}$$

whereas from [6, Lem. 7.13(b)] we obtain

$$\begin{aligned} \chi _{{\alpha ,2\ell +1}}\left( {z}\right) =z\mathfrak {h}_{{\alpha ,2\ell }}-\mathfrak {h}_{{\alpha ,2\ell +1}}-\mathfrak {h}_{{\alpha ,2\ell }}\left[ {\chi _{{\alpha ,2\ell -1}}\left( {z}\right) } \right] ^{\mathord {+}}\mathfrak {h}_{{\alpha ,2\ell }}. \end{aligned}$$

Remark 5.3 yields $\mathfrak {h}_{{\alpha ,2\ell }}\mathfrak {h}_{{\alpha ,2\ell -2}}^{\mathord {+}}\mathfrak {h}_{{\alpha ,2\ell -2}}=\mathfrak {h}_{{\alpha ,2\ell }}$. Remark 9.2 shows $\mathfrak {k}_{2\ell -1}=\mathfrak {h}_{{\alpha ,2\ell -2}}$ and $\mathfrak {k}_{2\ell +1}=\mathfrak {h}_{{\alpha ,2\ell }}$. Because of Lemma 12.12, we have furthermore $\left( {\mathfrak {h}_{{\alpha ,2\ell -2}}\left[ {\textbf{b}_{{{\mathord {\circ }},\ell -1}}\left( {z}\right) } \right] ^{-1}\textbf{b}_{{{\mathord {\circ }},\ell }}\left( {z}\right) }\right) ^{\mathord {+}}=\mathfrak {h}_{{\alpha ,2\ell -2}}^{\mathord {+}}\mathfrak {h}_{{\alpha ,2\ell -2}}\left[ {\textbf{b}_{{{\mathord {\circ }},\ell }}\left( {z}\right) } \right] ^{-1}\textbf{b}_{{{\mathord {\circ }},\ell -1}}\left( {z}\right) \mathfrak {h}_{{\alpha ,2\ell -2}}^{\mathord {+}}$. Using additionally (12.10) for $k=\ell -1$, we conclude

$$\begin{aligned}\begin{aligned}&\mathfrak {h}_{{\alpha ,2\ell }}\left[ {\textbf{b}_{{{\mathord {\circ }},\ell }}\left( {z}\right) } \right] ^{-1}\textbf{b}_{{{\mathord {\circ }},\ell -1}}\left( {z}\right) \mathfrak {k}_{2\ell -1}^{\mathord {+}}\mathfrak {k}_{2\ell +1}\\&=\mathfrak {h}_{{\alpha ,2\ell }}\mathfrak {h}_{{\alpha ,2\ell -2}}^{\mathord {+}}\mathfrak {h}_{{\alpha ,2\ell -2}}\left[ {\textbf{b}_{{{\mathord {\circ }},\ell }}\left( {z}\right) } \right] ^{-1}\textbf{b}_{{{\mathord {\circ }},\ell -1}}\left( {z}\right) \mathfrak {h}_{{\alpha ,2\ell -2}}^{\mathord {+}}\mathfrak {h}_{{\alpha ,2\ell }}\\&=\mathfrak {h}_{{\alpha ,2\ell }}\left( {\mathfrak {h}_{{\alpha ,2\ell -2}}\left[ {\textbf{b}_{{{\mathord {\circ }},\ell -1}}\left( {z}\right) } \right] ^{-1}\textbf{b}_{{{\mathord {\circ }},\ell }}\left( {z}\right) }\right) ^{\mathord {+}}\mathfrak {h}_{{\alpha ,2\ell }} =\mathfrak {h}_{{\alpha ,2\ell }}\left[ {\chi _{{\alpha ,2\ell -1}}\left( {z}\right) } \right] ^{\mathord {+}}\mathfrak {h}_{{\alpha ,2\ell }}. \end{aligned}\end{aligned}$$

Therefore, (12.10) follows for $k=\ell$. Thus, (12.10) is proved for each $k\in \mathbb {N}_0$ fulfilling $2k+2\le \kappa$ by mathematical induction. $\square$

If $\kappa \ge 1$ and $(s_j)_{j=0}^{\kappa }$ is a sequence of complex ${q\times q}$ matrices, then, in accordance with Notation 7.1, we denote in the sequel by $\left[ {(\mathfrak {a}_{{\alpha ,k}})_{k=0}^{\dot{\tau }},(\mathfrak {b}_{{\alpha ,k}})_{k=0}^{\dot{\tau }},(\mathfrak {c}_{{\alpha ,k}})_{k=0}^{\dot{\tau }},(\mathfrak {d}_{{\alpha ,k}})_{k=0}^{\dot{\tau }}} \right]$ the $\mathbb {R}$-QMP given by Definition 5.4 for the sequence $(a_j)_{j=0}^{\tau }$, where $\tau =\kappa -1$.

Lemma 12.14

Suppose $\kappa \ge 2$. Let $(s_j)_{j=0}^{\kappa }\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,\kappa ,\alpha }$, let $k\in \mathbb {N}_0$ be such that $2k+2\le \kappa$, let $\left( {\eta };{\theta }\right) \in \hat{\mathcal {P}}\left[ {\mathfrak {h}_{{\alpha ,2k}}} \right]$, and let $\mathcal {E}\in \mathscr {E}\left( {\eta ,\theta }\right)$. Then:

• (a) For all $z\in \mathbb {C}\backslash \left( {\mathbb {R}\cup \mathcal {E}}\right)$, we have

  $$\begin{aligned} \det \left( {\textbf{b}_{{{\mathord {\circ }},k}}\left( {z}\right) \mathfrak {h}_{{\alpha ,2k}}^{\mathord {+}}\eta \left( {z}\right) +\textbf{b}_{{{\mathord {\circ }},k+1}}\left( {z}\right) \theta \left( {z}\right) }\right)&\ne 0 \end{aligned}$$

  (12.11)

  and

  $$\begin{aligned} \det \left( {\mathfrak {b}_{{\alpha ,k}}\left( {z}\right) \mathfrak {h}_{{\alpha ,2k}}^{\mathord {+}}\eta \left( {z}\right) +\mathfrak {b}_{{\alpha ,k+1}}\left( {z}\right) \theta \left( {z}\right) }\right)&\ne 0. \end{aligned}$$

  (12.12)

• (b) For all $w{\in }\mathbb {C}\!\backslash \!\left( {\mathbb {R}\cup \mathcal {E}^{\mathord {\vee }}}\right)$, we have $\det \left( {\left[ {\eta \left( {\overline{w}}\right) } \right] ^*\mathfrak {h}_{{\alpha ,2k}}^{\mathord {+}}\textbf{d}_{{{\mathord {\circ }},k}}\left( {w}\right) \!+\!\left[ {\theta \left( {\overline{w}}\right) } \right] ^*\textbf{d}_{{{\mathord {\circ }},k\!+\!1}}\left( {w}\right) }\right) \ne 0$ and $\det \left( {\left[ {\eta \left( {\overline{w}}\right) } \right] ^*\mathfrak {h}_{{\alpha ,2k}}^{\mathord {+}}\mathfrak {d}_{{\alpha ,k}}\left( {w}\right) +\left[ {\theta \left( {\overline{w}}\right) } \right] ^*\mathfrak {d}_{{\alpha ,k+1}}\left( {w}\right) }\right) \ne 0$.

Proof

First observe that Remark 7.6 provides .

(a) We consider an arbitrary $z\in \mathbb {C}\backslash \left( {\mathbb {R}\cup \mathcal {E}}\right)$. The application of Lemma 6.5 to the sequence then yields $\left( {\Im z}\right) ^{-1}\Im \chi _{{\alpha ,2k+1}}\left( {z}\right) \succcurlyeq \mathfrak {h}_{{\alpha ,2k}}\succcurlyeq O_{{q\times q}}$. In particular, $\mathfrak {h}_{{\alpha ,2k}}\in \mathbb {C}_\succcurlyeq ^{{q\times q}}$. By virtue Notation 4.10, Definition 4.7, Remark 4.9(a), and Remark A.10, we can infer that $\left( {\eta };{\theta }\right)$ belongs to $\mathcal{P}\mathcal{R}_{q}\left( {\mathbb {C}\backslash \mathbb {R}}\right)$ and fulfills ${{\,\textrm{rank}\,}}\bigl [{\begin{matrix}\eta \left( {z}\right) \\ \theta \left( {z}\right) \end{matrix}}\bigr ]=q$ and $\left( {\Im z}\right) ^{-1}\Im \left( {\left[ {\theta \left( {z}\right) } \right] ^*\eta \left( {z}\right) }\right) \in \mathbb {C}_\succcurlyeq ^{{q\times q}}$ as well as $\mathfrak {h}_{{\alpha ,2k}}\mathfrak {h}_{{\alpha ,2k}}^{\mathord {+}}\eta \left( {z}\right) =\eta \left( {z}\right)$. First we are going to prove (12.11). Setting

$$\begin{aligned} B_{k}:=\textbf{b}_{{{\mathord {\circ }},k}}\left( {z}\right) ,\; D_{k}:=\textbf{d}_{{{\mathord {\circ }},k}}\left( {z}\right) ,\; B_{k+1}:=\textbf{b}_{{{\mathord {\circ }},k+1}}\left( {z}\right) , \text { and } D_{k+1}:=\textbf{d}_{{{\mathord {\circ }},k+1}}\left( {z}\right) , \nonumber \\ \end{aligned}$$

(12.13)

Lemma 12.11 yields

$$\begin{aligned} \det B_{k}&\ne 0,&\det D_{k}&\ne 0,&\det B_{k+1}&\ne 0,&\det D_{k+1}&\ne 0, \end{aligned}$$

(12.14)

and

$$\begin{aligned} \mathfrak {h}_{{\alpha ,2k}} B_{k}^{-1}B_{k+1} = D_{k+1} D_{k}^{-1}\mathfrak {h}_{{\alpha ,2k}}, \end{aligned}$$

(12.15)

whereas Lemma 12.13 provides

$$\begin{aligned} \chi _{{\alpha ,2k+1}}\left( {z}\right) =\mathfrak {h}_{{\alpha ,2k}} B_{k}^{-1}B_{k+1}. \end{aligned}$$

(12.16)

We consider an arbitrary $v\in \mathcal {N}\left( { B_{k}\mathfrak {h}_{{\alpha ,2k}}^{\mathord {+}}\eta \left( {z}\right) + B_{k+1}\theta \left( {z}\right) }\right)$. Then

$$\begin{aligned} B_{k}\mathfrak {h}_{{\alpha ,2k}}^{\mathord {+}}\eta \left( {z}\right) v+ B_{k+1}\theta \left( {z}\right) v =O_{{q\times 1}}. \end{aligned}$$

(12.17)

Regarding (12.14), we can multiply (12.17) from the left by $B_{k}^{-1}$ to obtain

$$\begin{aligned} \mathfrak {h}_{{\alpha ,2k}}^{\mathord {+}}\eta \left( {z}\right) v+ B_{k}^{-1}B_{k+1}\theta \left( {z}\right) v =O_{{q\times 1}}. \end{aligned}$$

(12.18)

Using (12.18) and $\mathfrak {h}_{{\alpha ,2k}}\mathfrak {h}_{{\alpha ,2k}}^{\mathord {+}}\eta \left( {z}\right) =\eta \left( {z}\right)$, we get

$$\begin{aligned} O_{{q\times 1}}= & {} \mathfrak {h}_{{\alpha ,2k}}\left[ {\mathfrak {h}_{{\alpha ,2k}}^{\mathord {+}}\eta \left( {z}\right) v+ B_{k}^{-1}B_{k+1}\theta \left( {z}\right) v} \right] \nonumber \\= & {} \eta \left( {z}\right) v+\mathfrak {h}_{{\alpha ,2k}} B_{k}^{-1}B_{k+1}\theta \left( {z}\right) v. \end{aligned}$$

(12.19)

Multiplying both sides of (12.19) from the left by $v^*\left[ {\theta \left( {z}\right) } \right] ^*$ and using (12.16), we obtain

$$\begin{aligned} 0= & {} v^*\left[ {\theta \left( {z}\right) } \right] ^*\eta \left( {z}\right) v+v^*\left[ {\theta \left( {z}\right) } \right] ^*\mathfrak {h}_{{\alpha ,2k}} B_{n}^{-1}B_{n+1}\theta \left( {z}\right) v \nonumber \\= & {} v^*\left[ {\theta \left( {z}\right) } \right] ^*\eta \left( {z}\right) v+\left[ {\theta \left( {z}\right) v} \right] ^*\chi _{{\alpha ,2k+1}}\left( {z}\right) \left[ {\theta \left( {z}\right) v} \right] . \end{aligned}$$

(12.20)

In view of $z\in \mathbb {C}\backslash \mathbb {R}$, multiplying both sides of (12.20) by $\left( {\Im z}\right) ^{-1}$, taking the imaginary part and regarding Remarks A.1 and A.2 as well as $\left( {\Im z}\right) ^{-1}\Im \chi _{{\alpha ,2k+1}}\left( {z}\right) \succcurlyeq \mathfrak {h}_{{\alpha ,2k}}\succcurlyeq O_{{q\times q}}$ and $\left( {\Im z}\right) ^{-1}\Im \left( {\left[ {\theta \left( {z}\right) } \right] ^*\eta \left( {z}\right) }\right) \in \mathbb {C}_\succcurlyeq ^{{q\times q}}$, we conclude

$$\begin{aligned} 0= & {} \Im \left( {v^*\left( {\left( {\Im z}\right) ^{-1}\left[ {\theta \left( {z}\right) } \right] ^*\eta \left( {z}\right) }\right) v+\left[ {\theta \left( {z}\right) v} \right] ^*\left[ {\left( {\Im z}\right) ^{-1}\chi _{{\alpha ,2k+1}}\left( {z}\right) } \right] \left[ {\theta \left( {z}\right) v} \right] }\right) \nonumber \\= & {} v^*\left[ {\left( {\Im z}\right) ^{-1}\Im \left( {\left[ {\theta \left( {z}\right) } \right] ^*\eta \left( {z}\right) }\right) } \right] v+\left[ {\theta \left( {z}\right) v} \right] ^*\left[ {\left( {\Im z}\right) ^{-1}\Im \chi _{{\alpha ,2k+1}}\left( {z}\right) } \right] \left[ {\theta \left( {z}\right) v} \right] \nonumber \\\ge & {} v^*\left[ {\left( {\Im z}\right) ^{-1}\Im \left( {\left[ {\theta \left( {z}\right) } \right] ^*\eta \left( {z}\right) }\right) } \right] v+\left[ {\theta \left( {z}\right) v} \right] ^*\mathfrak {h}_{{\alpha ,2 k }}\left[ {\theta \left( {z}\right) v} \right] \nonumber \\\ge & {} \left[ {\theta \left( {z}\right) v} \right] ^*\mathfrak {h}_{{\alpha ,2 k }}\left[ {\theta \left( {z}\right) v} \right] \ge 0. \end{aligned}$$

(12.21)

Consequently, we get $\left[ {\theta \left( {z}\right) v} \right] ^*\mathfrak {h}_{{\alpha ,2 k }}\left[ {\theta \left( {z}\right) v} \right] =0$ and, because of $\mathfrak {h}_{{\alpha ,2k}}\in \mathbb {C}_\succcurlyeq ^{{q\times q}}$, then $\mathfrak {h}_{{\alpha ,2k}}\theta \left( {z}\right) v=O_{{q\times 1}}$ follows. Using additionally (12.19) and (12.15), we conclude

$$\begin{aligned} O_{{q\times 1}}= & {} \eta \left( {z}\right) v+\mathfrak {h}_{{\alpha ,2 k }} B_{k}^{-1}B_{ k +1}\theta \left( {z}\right) v \nonumber \\= & {} \eta \left( {z}\right) v+ D_{ k +1} D_{k}^{-1}\mathfrak {h}_{{\alpha ,2 k }}\theta \left( {z}\right) v =\eta \left( {z}\right) v. \end{aligned}$$

(12.22)

Combining (12.17) and (12.22) yields $B_{k+1}\theta \left( {z}\right) v=O_{{q\times 1}}$ and, thus, (12.14) provides $\theta \left( {z}\right) v=O_{{q\times 1}}$. Taking additionally into account (12.22) and ${{\,\textrm{rank}\,}}\bigl [{\begin{matrix}\eta \left( {z}\right) \\ \theta \left( {z}\right) \end{matrix}}\bigr ]=q$, we infer $v=O_{{q\times 1}}$. Hence,

$$\begin{aligned} \mathcal {N}\left( { B_{k}\mathfrak {h}_{{\alpha ,2k}}^{\mathord {+}}\eta \left( {z}\right) + B_{k+1}\theta \left( {z}\right) }\right) \subseteq \left\{ {O_{{q\times 1}}} \right\} . \end{aligned}$$

(12.23)

In view of (12.13), therefore (12.11) is checked. In order to prove (12.12), we now set

$$\begin{aligned} B_{k}:=\mathfrak {b}_{{\alpha ,k}}\left( {z}\right) ,\; D_{k}:=\mathfrak {d}_{{\alpha ,k}}\left( {z}\right) ,\; B_{ k +1}:=\mathfrak {b}_{{\alpha , k +1}}\left( {z}\right) , \text { and } D_{ k +1}:=\mathfrak {d}_{{\alpha , k +1}}\left( {z}\right) .\nonumber \\ \end{aligned}$$

(12.24)

Regarding , we can then use Lemma 5.9 to obtain (12.14) and Remark 6.2 to get (12.16) and (12.15). We consider an arbitrary $v\in \mathcal {N}\left( { B_{k}\mathfrak {h}_{{\alpha ,2k}}^{\mathord {+}}\eta \left( {z}\right) + B_{k+1}\theta \left( {z}\right) }\right)$. Then (12.17) is fulfilled. Regarding (12.14), we can multiply (12.17) from the left by $B_{k}^{-1}$ to obtain (12.18). Using (12.18) and $\mathfrak {h}_{{\alpha ,2k}}\mathfrak {h}_{{\alpha ,2k}}^{\mathord {+}}\eta \left( {z}\right) =\eta \left( {z}\right)$, we again get (12.19). Multiplying both sides of (12.19) from the left by $v^*\left[ {\theta \left( {z}\right) } \right] ^*$ and using (12.16), we obtain (12.20). In view of $z\in \mathbb {C}\backslash \mathbb {R}$, multiplying both sides of (12.20) by $\left( {\Im z}\right) ^{-1}$, taking the imaginary part and regarding Remarks A.1 and A.2 as well as $\left( {\Im z}\right) ^{-1}\Im \chi _{{\alpha ,2k+1}}\left( {z}\right) \succcurlyeq \mathfrak {h}_{{\alpha ,2k}}\succcurlyeq O_{{q\times q}}$ and $\left( {\Im z}\right) ^{-1}\Im \left( {\left[ {\theta \left( {z}\right) } \right] ^*\eta \left( {z}\right) }\right) \in \mathbb {C}_\succcurlyeq ^{{q\times q}}$, we conclude that (12.21) is valid. Consequently, we get $\left[ {\theta \left( {z}\right) v} \right] ^*\mathfrak {h}_{{\alpha ,2 k }}\left[ {\theta \left( {z}\right) v} \right] =0$ and, because of $\mathfrak {h}_{{\alpha ,2k}}\in \mathbb {C}_\succcurlyeq ^{{q\times q}}$, then $\mathfrak {h}_{{\alpha ,2k}}\theta \left( {z}\right) v=O_{{q\times 1}}$ follows. Using additionally (12.19) and (12.15), we conclude (12.22). Combining (12.17) and (12.22), we obtain $B_{k+1}\theta \left( {z}\right) v=O_{{q\times 1}}$ and, thus, (12.14) justifies that $\theta \left( {z}\right) v=O_{{q\times 1}}$. Taking additionally into account (12.22) and regarding ${{\,\textrm{rank}\,}}\bigl [{\begin{matrix}\eta \left( {z}\right) \\ \theta \left( {z}\right) \end{matrix}}\bigr ]=q$, we infer $v=O_{{q\times 1}}$. Consequently, (12.23) is checked. From (12.24) we see that (12.12) is valid.

(b) Regarding , Remark 5.3 yields $\mathfrak {h}_{{\alpha ,2k}}^*=\mathfrak {h}_{{\alpha ,2k}}$. Thus, using Remark A.12, we can infer $\left( {\mathfrak {h}_{{\alpha ,2k}}^{\mathord {+}}}\right) ^*=\mathfrak {h}_{{\alpha ,2k}}^{\mathord {+}}$. We consider an arbitrary $w\in \mathbb {C}\backslash \left( {\mathbb {R}\cup \mathcal {E}^{\mathord {\vee }}}\right)$. In view of (2.1), then $\overline{w}\in \mathbb {C}\backslash \left( {\mathbb {R}\cup \mathcal {E}}\right)$. Remark 9.3 shows that $(s_j)_{j=0}^{\kappa }$ is a sequence of Hermitian matrices. Hence, we can apply Remark 12.2 to obtain furthermore $\left[ {\textbf{b}_{{{\mathord {\circ }},j}}\left( {\overline{w}}\right) } \right] ^*=\textbf{d}_{{{\mathord {\circ }},j}}\left( {w}\right)$ for each $j\in \left\{ {k,k+1} \right\}$. Consequently, we get $\left[ {\textbf{b}_{{{\mathord {\circ }},k}}\left( {\overline{w}}\right) \mathfrak {h}_{{\alpha ,2k}}^{\mathord {+}}\eta \left( {\overline{w}}\right) +\textbf{b}_{{{\mathord {\circ }},k+1}}\left( {\overline{w}}\right) \theta \left( {\overline{w}}\right) } \right] ^*=\left[ {\eta \left( {\overline{w}}\right) } \right] ^*\mathfrak {h}_{{\alpha ,2k}}^{\mathord {+}}\textbf{d}_{{{\mathord {\circ }},k}}\left( {w}\right) +\left[ {\theta \left( {\overline{w}}\right) } \right] ^*\textbf{d}_{{{\mathord {\circ }},k+1}}\left( {w}\right)$. Since part (a) implies $\det \left( {\textbf{b}_{{{\mathord {\circ }},k}}\left( {\overline{w}}\right) \mathfrak {h}_{{\alpha ,2k}}^{\mathord {+}}\eta \left( {\overline{w}}\right) \!+\!\textbf{b}_{{{\mathord {\circ }},k+1}}\left( {\overline{w}}\right) \theta \left( {\overline{w}}\right) }\right) \ne 0$, then $\det \left( {\left[ {\eta \left( {\overline{w}}\right) } \right] ^*\mathfrak {h}_{{\alpha ,2k}}^{\mathord {+}}\textbf{d}_{{{\mathord {\circ }},k}}\left( {w}\right) +\left[ {\theta \left( {\overline{w}}\right) } \right] ^*\textbf{d}_{{{\mathord {\circ }},k+1}}\left( {w}\right) }\right) \ne 0$ follows. By virtue of $\alpha \in \mathbb {R}$ and (7.1), we see that is a sequence of Hermitian matrices, as well. Thus, we can apply Remark 5.8 to get $\left[ {\mathfrak {b}_{{\alpha ,j}}\left( {\overline{w}}\right) } \right] ^*=\mathfrak {d}_{{\alpha ,j}}\left( {w}\right)$ for each $j\in \left\{ {k,k+1} \right\}$. As above, we can infer then analogously $\det \left( {\left[ {\eta \left( {\overline{w}}\right) } \right] ^*\mathfrak {h}_{{\alpha ,2k}}^{\mathord {+}}\mathfrak {d}_{{\alpha ,k}}\left( {w}\right) +\left[ {\theta \left( {\overline{w}}\right) } \right] ^*\mathfrak {d}_{{\alpha ,k+1}}\left( {w}\right) }\right) \ne 0$. $\square$

Lemma 12.15

Suppose $\kappa \ge 2$. Let $(s_j)_{j=0}^{\kappa }\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,\kappa ,\alpha }$. For all $k \in \mathbb {N}_0$ such that $2 k +2\le \kappa$, then

$$\begin{aligned} \begin{bmatrix}\mathfrak {c}_{{\alpha ,k}}&{}\mathfrak {d}_{{\alpha ,k}}\\ \mathfrak {c}_{{\alpha , k +1}}&{}\mathfrak {d}_{{\alpha , k +1}}\end{bmatrix} \left( {-\textrm{i}\tilde{J}_{q}}\right) \begin{bmatrix}\textbf{a}_{{{\mathord {\circ }},k}}&{}\textbf{a}_{{{\mathord {\circ }}, k +1}}\\ \textbf{b}_{{{\mathord {\circ }},k}}&{}\textbf{b}_{{{\mathord {\circ }}, k +1}}\end{bmatrix}&=\begin{bmatrix} O_{{q\times q}}&{}\mathfrak {h}_{{\alpha ,2 k }}\\ -\mathfrak {h}_{{\alpha ,2 k }}&{}O_{{q\times q}}\end{bmatrix}, \end{aligned}$$

(12.25)

$$\begin{aligned} \begin{bmatrix}\textbf{c}_{{{\mathord {\circ }},k}}&{}\textbf{d}_{{{\mathord {\circ }},k}}\\ \textbf{c}_{{{\mathord {\circ }}, k +1}}&{}\textbf{d}_{{{\mathord {\circ }}, k +1}}\end{bmatrix} \left( {-\textrm{i}\tilde{J}_{q}}\right) \begin{bmatrix}\mathfrak {a}_{{\alpha ,k}}&{}\mathfrak {a}_{{\alpha , k +1}}\\ \mathfrak {b}_{{\alpha ,k}}&{}\mathfrak {b}_{{\alpha , k +1}}\end{bmatrix}&=\begin{bmatrix} O_{{q\times q}}&{}\mathfrak {h}_{{\alpha ,2 k }}\\ -\mathfrak {h}_{{\alpha ,2 k }}&{}O_{{q\times q}}\end{bmatrix}, \end{aligned}$$

(12.26)

$$\begin{aligned} \begin{bmatrix}\mathfrak {c}_{{\alpha ,k}}&{}\mathfrak {d}_{{\alpha ,k}}\\ \mathfrak {c}_{{\alpha , k +1}}&{}\mathfrak {d}_{{\alpha , k +1}}\end{bmatrix} \left( {-\textrm{i}\tilde{J}_{q}}\right) \begin{bmatrix}\mathfrak {a}_{{\alpha ,k}}&{}\mathfrak {a}_{{\alpha , k +1}}\\ \mathfrak {b}_{{\alpha ,k}}&{}\mathfrak {b}_{{\alpha , k +1}}\end{bmatrix}&=\begin{bmatrix} O_{{q\times q}}&{}\mathfrak {h}_{{\alpha ,2 k }}\\ -\mathfrak {h}_{{\alpha ,2 k }}&{}O_{{q\times q}}\end{bmatrix}, \end{aligned}$$

(12.27)

and

$$\begin{aligned} \begin{bmatrix}\textbf{c}_{{{\mathord {\circ }},k}}&{}\textbf{d}_{{{\mathord {\circ }},k}}\\ \textbf{c}_{{{\mathord {\circ }}, k +1}}&{}\textbf{d}_{{{\mathord {\circ }}, k +1}}\end{bmatrix} \left( {-\textrm{i}\tilde{J}_{q}}\right) \begin{bmatrix}\textbf{a}_{{{\mathord {\circ }},k}}&{}\textbf{a}_{{{\mathord {\circ }}, k +1}}\\ \textbf{b}_{{{\mathord {\circ }},k}}&{}\textbf{b}_{{{\mathord {\circ }}, k +1}}\end{bmatrix}&=\begin{bmatrix} O_{{q\times q}}&{}\mathfrak {h}_{{\alpha ,2 k }}\\ -\mathfrak {h}_{{\alpha ,2 k }}&{}O_{{q\times q}}\end{bmatrix}. \end{aligned}$$

(12.28)

Proof

Remark 9.2 yields (9.2), whereas Lemma 9.6 provides (9.6). According to Remark 9.3, we have furthermore (9.3). Remark 7.6 yields . Thus, Remark 5.3 provides

$$\begin{aligned} \mathfrak {h}_{{\alpha ,2k-2}}\mathfrak {h}_{{\alpha ,2k-2}}^{\mathord {+}}\mathfrak {h}_{{\alpha ,2k-1}}&=\mathfrak {h}_{{\alpha ,2k-1}}{} & {} \text {and}&\mathfrak {h}_{{\alpha ,2k-1}}\mathfrak {h}_{{\alpha ,2k-2}}^{\mathord {+}}\mathfrak {h}_{{\alpha ,2k-2}}&=\mathfrak {h}_{{\alpha ,2k-1}} \end{aligned}$$

(12.29)

for all $k\in \mathbb {N}$ with $2k-1\le \kappa -1$ as well as

$$\begin{aligned} \mathfrak {h}_{{\alpha ,2k-2}}\mathfrak {h}_{{\alpha ,2k-2}}^{\mathord {+}}\mathfrak {h}_{{\alpha ,2k}}&=\mathfrak {h}_{{\alpha ,2k}}{} & {} \text {and}&\mathfrak {h}_{{\alpha ,2k}}\mathfrak {h}_{{\alpha ,2k-2}}^{\mathord {+}}\mathfrak {h}_{{\alpha ,2k-2}}&=\mathfrak {h}_{{\alpha ,2k}} \end{aligned}$$

(12.30)

for all $k\in \mathbb {N}$ with $2k\le \kappa -1$. We proceed by mathematical induction. Using (4.1), Definition 5.4 for , Notation 5.5, Lemmata 12.8 and 12.9, and (9.2), we can infer

where $R_0:=-\mathfrak {h}_{{\alpha ,0}}\left[ {\left( {\epsilon -\alpha }\right) I_{q}-\left( {\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{1}+\mathfrak {k}_{1}^{\mathord {+}}\mathfrak {k}_{2}}\right) } \right] +\left( {\epsilon I_{q}-\mathfrak {h}_{{\alpha ,1}}\mathfrak {h}_{{\alpha ,0}}^{\mathord {+}}}\right) \mathfrak {k}_{1}$. Taking into account (9.2), (9.3), (12.29), and (9.6), we can conclude

$$\begin{aligned}\begin{aligned} R_0&=-\mathfrak {k}_{1}\left[ {\left( {\epsilon -\alpha }\right) I_{q}-\left( {\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{1}+\mathfrak {k}_{1}^{\mathord {+}}\mathfrak {k}_{2}}\right) } \right] +\left( {\epsilon I_{q}-\mathfrak {h}_{{\alpha ,1}}\mathfrak {h}_{{\alpha ,0}}^{\mathord {+}}}\right) \mathfrak {h}_{{\alpha ,0}}\\&=-\epsilon \mathfrak {k}_{1}+\alpha \mathfrak {k}_{1}+\mathfrak {k}_{1}\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{1}+\mathfrak {k}_{1}\mathfrak {k}_{1}^{\mathord {+}}\mathfrak {k}_{2}+\epsilon \mathfrak {h}_{{\alpha ,0}}-\mathfrak {h}_{{\alpha ,1}}\mathfrak {h}_{{\alpha ,0}}^{\mathord {+}}\mathfrak {h}_{{\alpha ,0}}\\&=-\epsilon \mathfrak {h}_{{\alpha ,0}}+\alpha \mathfrak {k}_{1}+\mathfrak {k}_{1}\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{1}+\mathfrak {k}_{2}+\epsilon \mathfrak {h}_{{\alpha ,0}}-\mathfrak {h}_{{\alpha ,1}} =O_{{q\times q}}. \end{aligned} \end{aligned}$$

Consequently, (12.25) holds true for $k=0$. Analogously the same arguments can be used to check that (12.26) holds true for $k=0$. Similarly, applying (4.1), Definition 5.4 for , Notation 5.5, and (12.29), we can conclude that (12.27) holds true for $k=0$, whereas (4.1), Lemmata 12.8 and 12.9, (9.2), and (9.3) yield that (12.28) holds true for $k=0$. Now suppose $\kappa \ge 4$. Then, we have already shown that there exists an integer $\ell \in \mathbb {N}$ fulfilling $2\ell +2\le \kappa$ such that (12.25)–(12.28) hold true for $k=\ell -1$. Using Remark 5.6 for and (12.25) for $k=\ell -1$, we get

Taking into account Remark 12.10, (9.2), and (12.30), then

follows, where

$$\begin{aligned}{} & {} R_\ell :=\left( {\epsilon I_{q}-\mathfrak {h}_{{\alpha ,2\ell +1}}\mathfrak {h}_{{\alpha ,2\ell }}^{\mathord {+}}}\right) \mathfrak {h}_{{\alpha ,2\ell -2}}\mathfrak {k}_{2\ell -1}^{\mathord {+}}\mathfrak {k}_{2\ell +1}\\{} & {} \quad -\mathfrak {h}_{{\alpha ,2\ell }}\mathfrak {h}_{{\alpha ,2\ell -2}}^{\mathord {+}}\mathfrak {h}_{{\alpha ,2\ell -2}}\left[ {\left( {\epsilon -\alpha }\right) I_{q}-\left( {\mathfrak {k}_{2\ell }^{\mathord {+}}\mathfrak {k}_{2\ell +1}+\mathfrak {k}_{2\ell +1}^{\mathord {+}}\mathfrak {k}_{2\ell +2}}\right) } \right] . \end{aligned}$$

Using (9.2), (12.30), (9.3), (12.29), and (9.6), we can conclude

$$\begin{aligned}\begin{aligned} R_\ell&=\left( {\epsilon I_{q}-\mathfrak {h}_{{\alpha ,2\ell +1}}\mathfrak {h}_{{\alpha ,2\ell }}^{\mathord {+}}}\right) \mathfrak {h}_{{\alpha ,2\ell -2}}\mathfrak {h}_{{\alpha ,2\ell -2}}^{\mathord {+}}\mathfrak {h}_{{\alpha ,2\ell }}\\&\qquad -\mathfrak {k}_{2\ell +1}\mathfrak {k}_{2\ell -1}^{\mathord {+}}\mathfrak {k}_{2\ell -1}\left[ {\left( {\epsilon -\alpha }\right) I_{q}-\left( {\mathfrak {k}_{2\ell }^{\mathord {+}}\mathfrak {k}_{2\ell +1}+\mathfrak {k}_{2\ell +1}^{\mathord {+}}\mathfrak {k}_{2\ell +2}}\right) } \right] \\&=\left( {\epsilon I_{q}-\mathfrak {h}_{{\alpha ,2\ell +1}}\mathfrak {h}_{{\alpha ,2\ell }}^{\mathord {+}}}\right) \mathfrak {h}_{{\alpha ,2\ell }}-\mathfrak {k}_{2\ell +1}\left[ {\left( {\epsilon -\alpha }\right) I_{q}-\left( {\mathfrak {k}_{2\ell }^{\mathord {+}}\mathfrak {k}_{2\ell +1}+\mathfrak {k}_{2\ell +1}^{\mathord {+}}\mathfrak {k}_{2\ell +2}}\right) } \right] \\&=\epsilon \mathfrak {h}_{{\alpha ,2\ell }}-\mathfrak {h}_{{\alpha ,2\ell +1}}\mathfrak {h}_{{\alpha ,2\ell }}^{\mathord {+}}\mathfrak {h}_{{\alpha ,2\ell }}\\&\qquad -\epsilon \mathfrak {k}_{2\ell +1}+\alpha \mathfrak {k}_{2\ell +1}+\mathfrak {k}_{2\ell +1}\mathfrak {k}_{2\ell }^{\mathord {+}}\mathfrak {k}_{2\ell +1}+\mathfrak {k}_{2\ell +1}\mathfrak {k}_{2\ell +1}^{\mathord {+}}\mathfrak {k}_{2\ell +2}\\&=\epsilon \mathfrak {h}_{{\alpha ,2\ell }}-\mathfrak {h}_{{\alpha ,2\ell +1}}-\epsilon \mathfrak {k}_{2\ell +1}+\alpha \mathfrak {k}_{2\ell +1}+\mathfrak {k}_{2\ell +1}\mathfrak {k}_{2\ell }^{\mathord {+}}\mathfrak {k}_{2\ell +1}+\mathfrak {k}_{2\ell +2} =O_{{q\times q}}. \end{aligned}\end{aligned}$$

Consequently, (12.25) holds true for $k=\ell$. Using Remark 12.10 and (12.26) for $k=\ell -1$ as well as Remark 5.6 for , (9.2), and (12.30), we get similarly

$$\begin{aligned} \begin{bmatrix}\textbf{c}_{{{\mathord {\circ }},\ell }}&{}\textbf{d}_{{{\mathord {\circ }},\ell }}\\ \textbf{c}_{{{\mathord {\circ }},\ell +1}}&{}\textbf{d}_{{{\mathord {\circ }},\ell +1}}\end{bmatrix}\left( {-\textrm{i}\tilde{J}_{q}}\right) \begin{bmatrix}\mathfrak {a}_{{\alpha ,\ell }}&{}\mathfrak {a}_{{\alpha ,\ell +1}}\\ \mathfrak {b}_{{\alpha ,\ell }}&{}\mathfrak {b}_{{\alpha ,\ell +1}}\end{bmatrix} =\begin{bmatrix}O_{{q\times q}}&{}\mathfrak {h}_{{\alpha ,2\ell }}\\ hpa{2\ell }&{}S_\ell \end{bmatrix}, \end{aligned}$$

where

$$\begin{aligned} S_\ell :=\left[ {\left( {\epsilon -\alpha }\right) I_{q}-\left( {\mathfrak {k}_{2\ell +1}\mathfrak {k}_{2\ell }^{\mathord {+}}+\mathfrak {k}_{2\ell +2}\mathfrak {k}_{2\ell +1}^{\mathord {+}}}\right) } \right] \mathfrak {h}_{{\alpha ,2\ell -2}}\mathfrak {h}_{{\alpha ,2\ell -2}}^{\mathord {+}}\mathfrak {h}_{{\alpha ,2\ell }}\\ -\mathfrak {k}_{2\ell +1}\mathfrak {k}_{2\ell -1}^{\mathord {+}}\mathfrak {h}_{{\alpha ,2\ell -2}}\left( {\epsilon I_{q}-\mathfrak {h}_{{\alpha ,2\ell }}^{\mathord {+}}\mathfrak {h}_{{\alpha ,2\ell +1}}}\right) . \end{aligned}$$

Applying (9.2), (9.3), (12.30), (12.29), and (9.6), we can conclude that $S_\ell =O_{{q\times q}}$. Consequently, (12.26) holds true for $k=\ell$. Using Remark 5.6 for , (12.27) for $k=\ell -1$, (12.30), and (12.29), we can conclude that (12.27) holds true for $k=\ell$, whereas Remark 12.10, (12.28) for $k=\ell -1$, (9.2), (12.30), (9.3) provide that (12.28) holds true for $k=\ell$. Thus, (12.25)–(12.28) are inductively proved for all $k\in \mathbb {N}_0$ fulfilling $2k+2\le \kappa$. $\square$

Proposition 12.16

Suppose $\kappa \ge 2$. Let $(s_j)_{j=0}^{\kappa }\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,\kappa ,\alpha }$, let $k \in \mathbb {N}_0$ be such that $2 k +2\le \kappa$, and let $\left( {\eta };{\theta }\right) \in \hat{\mathcal {P}}\left[ {\mathfrak {h}_{{\alpha ,2 k }}} \right]$. Let $\mathcal {E}\in \mathscr {E}\left( {\eta ,\theta }\right)$ and let $\hat{\mathcal {E}}:=\mathcal {E}\cup \mathcal {E}^{\mathord {\vee }}$. For all $z\in \mathbb {C}\backslash \left( {\mathbb {R}\cup \hat{\mathcal {E}}}\right)$, then

$$\begin{aligned} \det \left( {\mathfrak {b}_{{\alpha ,k}}\left( {z}\right) \mathfrak {h}_{{\alpha ,2 k }}^{\mathord {+}}\eta \left( {z}\right) +\mathfrak {b}_{{\alpha , k +1}}\left( {z}\right) \theta \left( {z}\right) }\right)&\ne 0,\end{aligned}$$

(12.31)

$$\begin{aligned} \det \left( {\textbf{b}_{{{\mathord {\circ }},k}}\left( {z}\right) \mathfrak {h}_{{\alpha ,2 k }}^{\mathord {+}}\eta \left( {z}\right) +\textbf{b}_{{{\mathord {\circ }}, k +1}}\left( {z}\right) \theta \left( {z}\right) }\right)&\ne 0,\end{aligned}$$

(12.32)

$$\begin{aligned} \det \left( {\left[ {\eta \left( {\overline{z}}\right) } \right] ^*\mathfrak {h}_{{\alpha ,2 k }}^{\mathord {+}}\mathfrak {d}_{{\alpha ,k}}\left( {z}\right) +\left[ {\theta \left( {\overline{z}}\right) } \right] ^*\mathfrak {d}_{{\alpha , k +1}}\left( {z}\right) }\right)&\ne 0 \end{aligned}$$

(12.33)

and

$$\begin{aligned} \det \left( {\left[ {\eta \left( {\overline{z}}\right) } \right] ^*\mathfrak {h}_{{\alpha ,2 k }}^{\mathord {+}}\textbf{d}_{{{\mathord {\circ }},k}}\left( {z}\right) +\left[ {\theta \left( {\overline{z}}\right) } \right] ^*\textbf{d}_{{{\mathord {\circ }}, k +1}}\left( {z}\right) }\right)&\ne 0, \end{aligned}$$

(12.34)

hold true and the matrix-valued function $G:=-\left( {\mathfrak {a}_{{\alpha ,k}}\mathfrak {h}_{{\alpha ,2 k }}^{\mathord {+}}\eta +\mathfrak {a}_{{\alpha , k +1}}\theta }\right) \left( {\mathfrak {b}_{{\alpha ,k}}\mathfrak {h}_{{\alpha ,2 k }}^{\mathord {+}}\eta +\mathfrak {b}_{{\alpha , k +1}}\theta }\right) ^{-1}$ admits, for all $z\in \mathbb {C}\backslash \left( {\mathbb {R}\cup \hat{\mathcal {E}}}\right)$, the representations

$$\begin{aligned} G\left( {z}\right)&=-\left[ {\mathfrak {a}_{{\alpha ,k}}\left( {z}\right) \mathfrak {h}_{{\alpha ,2 k }}^{\mathord {+}}\eta \left( {z}\right) +\mathfrak {a}_{{\alpha , k +1}}\left( {z}\right) \theta \left( {z}\right) } \right] \\&\qquad \times \left[ {\mathfrak {b}_{{\alpha ,k}}\left( {z}\right) \mathfrak {h}_{{\alpha ,2 k }}^{\mathord {+}}\eta \left( {z}\right) +\mathfrak {b}_{{\alpha , k +1}}\left( {z}\right) \theta \left( {z}\right) } \right] ^{-1},\\ G\left( {z}\right)&=-\left( {\left[ {\eta \left( {\overline{z}}\right) } \right] ^*\mathfrak {h}_{{\alpha ,2 k }}^{\mathord {+}}\mathfrak {d}_{{\alpha ,k}}\left( {z}\right) +\left[ {\theta \left( {\overline{z}}\right) } \right] ^*\mathfrak {d}_{{\alpha , k +1}}\left( {z}\right) }\right) ^{-1}\\&\qquad \times \left( {\left[ {\eta \left( {\overline{z}}\right) } \right] ^*\mathfrak {h}_{{\alpha ,2 k }}^{\mathord {+}}\mathfrak {c}_{{\alpha ,k}}\left( {z}\right) +\left[ {\theta \left( {\overline{z}}\right) } \right] ^*\mathfrak {c}_{{\alpha , k +1}}\left( {z}\right) }\right) ,\\ G\left( {z}\right)&=-\left[ {\textbf{a}_{{{\mathord {\circ }},k}}\left( {z}\right) \mathfrak {h}_{{\alpha ,2 k }}^{\mathord {+}}\eta \left( {z}\right) +\textbf{a}_{{{\mathord {\circ }}, k +1}}\left( {z}\right) \theta \left( {z}\right) } \right] \\&\qquad \times \left[ {\textbf{b}_{{{\mathord {\circ }},k}}\left( {z}\right) \mathfrak {h}_{{\alpha ,2 k }}^{\mathord {+}}\eta \left( {z}\right) +\textbf{b}_{{{\mathord {\circ }}, k +1}}\left( {z}\right) \theta \left( {z}\right) } \right] ^{-1}, \end{aligned}$$

and

$$\begin{aligned} G\left( {z}\right)&=-\left( {\left[ {\eta \left( {\overline{z}}\right) } \right] ^*\mathfrak {h}_{{\alpha ,2 k }}^{\mathord {+}}\textbf{d}_{{{\mathord {\circ }},k}}\left( {z}\right) +\left[ {\theta \left( {\overline{z}}\right) } \right] ^*\textbf{d}_{{{\mathord {\circ }}, k +1}}\left( {z}\right) }\right) ^{-1}\\&\qquad \times \left( {\left[ {\eta \left( {\overline{z}}\right) } \right] ^*\mathfrak {h}_{{\alpha ,2 k }}^{\mathord {+}}\textbf{c}_{{{\mathord {\circ }},k}}\left( {z}\right) +\left[ {\theta \left( {\overline{z}}\right) } \right] ^*\textbf{c}_{{{\mathord {\circ }}, k +1}}\left( {z}\right) }\right) . \end{aligned}$$

Proof

Let $z\in \mathbb {C}\backslash \left( {\mathbb {R}\cup \hat{\mathcal {E}}}\right)$. Then (12.31)–(12.34) follow immediately from Lemma 12.14. Since Remark 7.6 provides , we can apply Remark 5.3 to to obtain $\mathfrak {h}_{{\alpha ,2k}}^*=\mathfrak {h}_{{\alpha ,2k}}$. Thus, using Remark A.12, we conclude $\left( {\mathfrak {h}_{{\alpha ,2k}}^{\mathord {+}}}\right) ^*=\mathfrak {h}_{{\alpha ,2k}}^{\mathord {+}}$. According to Notation 4.10, the pair $\left( {\eta };{\theta }\right)$ belongs to $\mathcal{P}\mathcal{R}_{q}\left( {\mathbb {C}\backslash \mathbb {R}}\right)$ and fulfills $\mathbb {P}_{\mathcal {R}\left( {\mathfrak {h}_{{\alpha ,2k}}}\right) }\eta =\eta$. Remark 4.9(b) then yields $\left[ {\eta \left( {\overline{z}}\right) } \right] ^*\theta \left( {z}\right) =\left[ {\theta \left( {\overline{z}}\right) } \right] ^*\eta \left( {z}\right)$, whereas Remark A.10 provides $\mathfrak {h}_{{\alpha ,2k}}\mathfrak {h}_{{\alpha ,2k}}^{\mathord {+}}\eta =\eta$. Consequently,

(12.35)

Let

$$\begin{aligned} {\mathfrak {A}}&:=\mathfrak {a}_{{\alpha ,k}}\left( {z}\right) \mathfrak {h}_{{\alpha ,2k}}^{\mathord {+}}\eta \left( {z}\right) +\mathfrak {a}_{{\alpha ,k+1}}\left( {z}\right) \theta \left( {z}\right) ,\\ {\mathfrak {B}}&:=\mathfrak {b}_{{\alpha ,k}}\left( {z}\right) \mathfrak {h}_{{\alpha ,2k}}^{\mathord {+}}\eta \left( {z}\right) +\mathfrak {b}_{{\alpha ,k+1}}\left( {z}\right) \theta \left( {z}\right) ,\\ {\mathfrak {C}}&:=\left[ {\eta \left( {\overline{z}}\right) } \right] ^*\mathfrak {h}_{{\alpha ,2k}}^{\mathord {+}}\mathfrak {c}_{{\alpha ,k}}\left( {z}\right) +\left[ {\theta \left( {\overline{z}}\right) } \right] ^*\mathfrak {c}_{{\alpha ,k+1}}\left( {z}\right) ,\\ {\mathfrak {D}}&:=\left[ {\eta \left( {\overline{z}}\right) } \right] ^*\mathfrak {h}_{{\alpha ,2k}}^{\mathord {+}}\mathfrak {d}_{{\alpha ,k}}\left( {z}\right) +\left[ {\theta \left( {\overline{z}}\right) } \right] ^*\mathfrak {d}_{{\alpha ,k+1}}\left( {z}\right) \end{aligned}$$

as well as

$$\begin{aligned} {\textbf{A}}&:=\textbf{a}_{{{\mathord {\circ }},k}}\left( {z}\right) \mathfrak {h}_{{\alpha ,2k}}^{\mathord {+}}\eta \left( {z}\right) +\textbf{a}_{{{\mathord {\circ }},k+1}}\left( {z}\right) \theta \left( {z}\right) ,\\ {\textbf{B}}&:=\textbf{b}_{{{\mathord {\circ }},k}}\left( {z}\right) \mathfrak {h}_{{\alpha ,2k}}^{\mathord {+}}\eta \left( {z}\right) +\textbf{b}_{{{\mathord {\circ }},k+1}}\left( {z}\right) \theta \left( {z}\right) ,\\ {\textbf{C}}&:=\left[ {\eta \left( {\overline{z}}\right) } \right] ^*\mathfrak {h}_{{\alpha ,2k}}^{\mathord {+}}\textbf{c}_{{{\mathord {\circ }},k}}\left( {z}\right) +\left[ {\theta \left( {\overline{z}}\right) } \right] ^*\textbf{c}_{{{\mathord {\circ }},k+1}}\left( {z}\right) ,\\ {\textbf{D}}&:=\left[ {\eta \left( {\overline{z}}\right) } \right] ^*\mathfrak {h}_{{\alpha ,2k}}^{\mathord {+}}\textbf{d}_{{{\mathord {\circ }},k}}\left( {z}\right) +\left[ {\theta \left( {\overline{z}}\right) } \right] ^*\textbf{d}_{{{\mathord {\circ }},k+1}}\left( {z}\right) . \end{aligned}$$

According to Lemma 12.15, we have (12.25)–(12.28). By virtue of (4.1), (12.27), and (12.35), we obtain

Taking additionally into account (12.31) and (12.33), then ${\mathfrak {A}}{\mathfrak {B}}^{-1}={\mathfrak {D}}^{-1}{\mathfrak {C}}$ follows. Similarly, (4.1), (12.28), and (12.35) yield ${\textbf{D}}{\textbf{A}}-{\textbf{C}}{\textbf{B}}=O_{{q\times q}}$, which in view of (12.32) and (12.34) implies ${\textbf{A}}{\textbf{B}}^{-1}={\textbf{D}}^{-1}{\textbf{C}}$. Using (4.1), (12.25), (12.35), (12.32), and (12.33), in the same way we get ${\textbf{A}}{\textbf{B}}^{-1}={\mathfrak {D}}^{-1}{\mathfrak {C}}$. Summarizing, we infer ${\mathfrak {A}}{\mathfrak {B}}^{-1}={\mathfrak {D}}^{-1}{\mathfrak {C}}={\textbf{A}}{\textbf{B}}^{-1}={\textbf{D}}^{-1}{\textbf{C}}$, which completes the proof. $\square$

Corollary 12.17

Suppose $\kappa \ge 2$. Let $(s_j)_{j=0}^{\kappa }\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,\kappa ,\alpha }$, let $k\in \mathbb {N}_0$ be such that $2k+2\le \kappa$, let $\left( {\phi };{\psi }\right) \in \mathcal {P}\left[ {\mathfrak {h}_{{\alpha ,2k}}} \right]$, and let $\mathcal {D}\in \mathscr {D}\left( {\phi ,\psi }\right)$. For all $z\in \Pi _{\mathord {+}}\backslash \mathcal {D}$, then $\det \left( {\mathfrak {b}_{{\alpha ,k}}\left( {z}\right) \mathfrak {h}_{{\alpha ,2k}}^{\mathord {+}}\phi \left( {z}\right) +\mathfrak {b}_{{\alpha ,k+1}}\left( {z}\right) \psi \left( {z}\right) }\right) \ne 0$ and $\det \left( {\textbf{b}_{{{\mathord {\circ }},k}}\left( {z}\right) \mathfrak {h}_{{\alpha ,2k}}^{\mathord {+}}\phi \left( {z}\right) +\textbf{b}_{{{\mathord {\circ }},k+1}}\left( {z}\right) \psi \left( {z}\right) }\right) \ne 0$ as well as

$$\begin{aligned}{} & {} -\left[ {\mathfrak {a}_{{\alpha ,k}}\left( {z}\right) \mathfrak {h}_{{\alpha ,2k}}^{\mathord {+}}\phi \left( {z}\right) +\mathfrak {a}_{{\alpha ,k+1}}\left( {z}\right) \psi \left( {z}\right) } \right] \\{} & {} \qquad \times \left[ {\mathfrak {b}_{{\alpha ,k}}\left( {z}\right) \mathfrak {h}_{{\alpha ,2k}}^{\mathord {+}}\phi \left( {z}\right) +\mathfrak {b}_{{\alpha ,k+1}}\left( {z}\right) \psi \left( {z}\right) } \right] ^{-1}\\{} & {} \quad =-\left[ {\textbf{a}_{{{\mathord {\circ }},k}}\left( {z}\right) \mathfrak {h}_{{\alpha ,2k}}^{\mathord {+}}\phi \left( {z}\right) +\textbf{a}_{{{\mathord {\circ }},k+1}}\left( {z}\right) \psi \left( {z}\right) } \right] \\{} & {} \qquad \times \left[ {\textbf{b}_{{{\mathord {\circ }},k}}\left( {z}\right) \mathfrak {h}_{{\alpha ,2k}}^{\mathord {+}}\phi \left( {z}\right) +\textbf{b}_{{{\mathord {\circ }},k+1}}\left( {z}\right) \psi \left( {z}\right) } \right] ^{-1}. \end{aligned}$$

Proof

According to Notation 4.5, we have $\left( {\phi };{\psi }\right) \in \mathcal{P}\mathcal{R}_{q}\left( {\Pi _{\mathord {+}}}\right)$. Thus, Lemma 4.11(e) shows that $\left( {\phi _{\mathord {\diamond }}};{\psi _{\mathord {\diamond }}}\right)$ given by (4.4) belongs to $\mathcal{P}\mathcal{R}_{q}\left( {\mathbb {C}\backslash \mathbb {R}}\right)$ and that $\mathcal {E}:=\mathcal {D}\cup \mathcal {D}^{\mathord {\vee }}$ belongs to $\mathscr {E}\left( {\phi _{\mathord {\diamond }},\psi _{\mathord {\diamond }}}\right)$. We now consider an arbitrary $z\in \Pi _{\mathord {+}}\backslash \mathcal {D}$. In view of Definition 4.2, we have $\mathcal {D}\subseteq \Pi _{\mathord {+}}$. Regarding (2.1), thus $z\notin \mathcal {D}^{\mathord {\vee }}$. Consequently, $z\notin \mathcal {E}$. By virtue of (2.1), it is readily checked that $\mathcal {E}^{\mathord {\vee }}=\mathcal {E}$. Hence, $\hat{\mathcal {E}}:=\mathcal {E}\cup \mathcal {E}^{\mathord {\vee }}$ fulfills $\hat{\mathcal {E}}=\mathcal {E}$. Summarizing, we can conclude $z\in \mathbb {C}\backslash \left( {\mathbb {R}\cup \hat{\mathcal {E}}}\right)$. Since Remark 7.6 provides , we can apply Remark 5.3 to to obtain $\mathfrak {h}_{{\alpha ,2k}}\in \mathbb {C}_\textrm{H}^{{q\times q}}$. From Lemma 4.12 we can infer then $\left( {\phi _{\mathord {\diamond }}};{\psi _{\mathord {\diamond }}}\right) \in \hat{\mathcal {P}}\left[ {\mathfrak {h}_{{\alpha ,2k}}} \right]$. Taking additionally into account $z\in \mathbb {C}\backslash \left( {\mathbb {R}\cup \hat{\mathcal {E}}}\right)$, the application of Proposition 12.16 then yields $\det \left( {\mathfrak {b}_{{\alpha ,k}}\left( {z}\right) \mathfrak {h}_{{\alpha ,2k}}^{\mathord {+}}\phi _{\mathord {\diamond }}\left( {z}\right) +\mathfrak {b}_{{\alpha ,k+1}}\left( {z}\right) \psi _{\mathord {\diamond }}\left( {z}\right) }\right) \ne 0$ and $\det \left( {\textbf{b}_{{{\mathord {\circ }},k}}\left( {z}\right) \mathfrak {h}_{{\alpha ,2k}}^{\mathord {+}}\phi _{\mathord {\diamond }}\left( {z}\right) +\textbf{b}_{{{\mathord {\circ }},k+1}}\left( {z}\right) \psi _{\mathord {\diamond }}\left( {z}\right) }\right) \ne 0$ and that $G:=-\left( {\mathfrak {a}_{{\alpha ,k}}\mathfrak {h}_{{\alpha ,2 k }}^{\mathord {+}}\phi _{\mathord {\diamond }}+\mathfrak {a}_{{\alpha , k +1}}\psi _{\mathord {\diamond }}}\right) \left( {\mathfrak {b}_{{\alpha ,k}}\mathfrak {h}_{{\alpha ,2 k }}^{\mathord {+}}\phi _{\mathord {\diamond }}+\mathfrak {b}_{{\alpha , k +1}}\psi _{\mathord {\diamond }}}\right) ^{-1}$ fulfills $G\left( {z}\right) =-\left[ {\mathfrak {a}_{{\alpha ,k}}\left( {z}\right) \mathfrak {h}_{{\alpha ,2k}}^{\mathord {+}}\phi _{\mathord {\diamond }}\left( {z}\right) +\mathfrak {a}_{{\alpha ,k+1}}\left( {z}\right) \psi _{\mathord {\diamond }}\left( {z}\right) } \right] \left[ {\mathfrak {b}_{{\alpha ,k}}\left( {z}\right) \mathfrak {h}_{{\alpha ,2k}}^{\mathord {+}}\phi _{\mathord {\diamond }}\left( {z}\right) +\mathfrak {b}_{{\alpha ,k+1}}\left( {z}\right) \psi _{\mathord {\diamond }}\left( {z}\right) } \right] ^{-1}$ and $G\left( {z}\right) =-\left[ {\textbf{a}_{{{\mathord {\circ }},k}}\left( {z}\right) \mathfrak {h}_{{\alpha ,2k}}^{\mathord {+}}\phi _{\mathord {\diamond }}\left( {z}\right) +\textbf{a}_{{{\mathord {\circ }},k+1}}\left( {z}\right) \psi _{\mathord {\diamond }}\left( {z}\right) } \right] \left[ {\textbf{b}_{{{\mathord {\circ }},k}}\left( {z}\right) \mathfrak {h}_{{\alpha ,2k}}^{\mathord {+}}\phi _{\mathord {\diamond }}\left( {z}\right) +\textbf{b}_{{{\mathord {\circ }},k+1}}\left( {z}\right) \psi _{\mathord {\diamond }}\left( {z}\right) } \right] ^{-1}$. Since Lemma 4.11(d) provides $\phi _{\mathord {\diamond }}\left( {z}\right) =\phi \left( {z}\right)$ and $\psi _{\mathord {\diamond }}\left( {z}\right) =\psi \left( {z}\right)$, the assertions follow. $\square$

13 Two Associated Matrix Balls

In this section, we are going to prove that the values of the Stieltjes transforms of solutions of the truncated matricial Stieltjes power moment problem belong to the intersection of two distinguished matrix balls.

If $(s_j)_{j=0}^{\kappa }\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,\kappa }$ and $z\in \mathbb {C}\backslash \mathbb {R}$, then, in view of Notations 6.6 and 6.7, let

$$\begin{aligned} \mathscr {K}_{m}\left( {z}\right) :=\mathfrak {K}\left( {\mathscr {C}_{m}\left( {z}\right) ;\left( {z-\overline{z}}\right) ^{-1}\mathscr {A}_{m}\left( {z}\right) ,\mathscr {B}_{m}\left( {z}\right) }\right) \end{aligned}$$

(13.1)

for each $m\in \mathbb {Z}_{0,\kappa }$.

Lemma 13.1

Let $n\in \mathbb {N}_0$, let $(s_j)_{j=0}^{2n}\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,2n,\alpha }$, let $F\in {\mathcal {S}_{0,q}\left[ {\mathbb {C}\backslash {[\alpha ,\infty )};(s_j)_{j=0}^{2n},\preccurlyeq } \right] }$, and let $w\in \Pi _{\mathord {+}}$. Then $(s_j)_{j=0}^{2n}\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,2n}$ and $F\left( {w}\right) \in \mathscr {K}_{2n}\left( {w}\right)$.

Proof

Lemma 7.5(a) yields $(s_j)_{j=0}^{2n}\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,2n}$, whereas Remark 7.8 shows $F\in \mathcal {S}_{0,q}\left( {\mathbb {C}\backslash {[\alpha ,\infty )}}\right)$ and that the ${[\alpha ,\infty )}$-spectral measure $\sigma$ of F belongs to ${\mathcal {M}^{q}_\succcurlyeq [{[\alpha ,\infty )};(s_j)_{j=0}^{2n},\preccurlyeq ]}$. In particular, (7.5) is valid and we have $\sigma \in \mathcal {M}_{\succcurlyeq ,2n}^{q}({[\alpha ,\infty )})$ as well as $\int _{{[\alpha ,\infty )}}x^{2n}\sigma \left( {\textrm{d}x}\right) \preccurlyeq s_{2n}$ and, in the case $n\ge 1$, for all $j\in \mathbb {Z}_{0,2n-1}$, moreover,

$$\begin{aligned} \int _{{[\alpha ,\infty )}}x^j\sigma \left( {\textrm{d}x}\right) =s_{j}. \end{aligned}$$

(13.2)

Remark B.3 and Remark B.4(b) then yield that ${\hat{\sigma }}:{\mathfrak {B}_{\mathbb {R}}}\rightarrow \mathbb {C}^{{q\times q}}$ defined by ${\hat{\sigma }}(B):=\sigma \left( {B\cap {[\alpha ,\infty )}}\right)$ belongs to $\mathcal {M}_{\succcurlyeq ,2n}^{q}(\mathbb {R})$ and fulfills ${{\,\textrm{Rstr}\,}}_{\mathfrak {B}_{{[\alpha ,\infty )}}}{{\hat{\sigma }}}=\sigma$ and ${\hat{\sigma }}\left( {\mathbb {R}\backslash {[\alpha ,\infty )}}\right) =O_{{q\times q}}$ as well as $\int _{{[\alpha ,\infty )}}x^j{\hat{\sigma }}\left( {\textrm{d}x}\right) =\int _{{[\alpha ,\infty )}}x^j\sigma \left( {\textrm{d}x}\right)$ for all $j\in \mathbb {Z}_{0,2n}$. Using Remark B.2, then $\int _\mathbb {R}x^j{\hat{\sigma }}\left( {\textrm{d}x}\right) =\int _{\mathbb {R}\backslash \left( {\mathbb {R}\backslash {[\alpha ,\infty )}}\right) }x^j{\hat{\sigma }}\left( {\textrm{d}x}\right) =\int _{{[\alpha ,\infty )}}x^j\sigma \left( {\textrm{d}x}\right)$ for all $j\in \mathbb {Z}_{0,2n}$ follows. Combining this with $\int _{{[\alpha ,\infty )}}x^{2n}\sigma \left( {\textrm{d}x}\right) \preccurlyeq s_{2n}$ and (13.2), we get ${\hat{\sigma }}\in {\mathcal {M}^{q}_\succcurlyeq [\mathbb {R};(s_j)_{j=0}^{2n},\preccurlyeq ]}$. Taking Theorem 3.2 into account, we see that $f:\Pi _{\mathord {+}}\rightarrow \mathbb {C}^{{q\times q}}$ given by $f\left( {z}\right) :=\int _\mathbb {R}\left( {x-z}\right) ^{-1}{\hat{\sigma }}\left( {\textrm{d}x}\right)$ is a well-defined matrix-valued function belonging to $\mathcal {R}_{0,q}\left( {\Pi _{\mathord {+}}}\right)$ and that ${\hat{\sigma }}$ is the $\mathbb {R}$-spectral measure of f. Thus, in view of ${\hat{\sigma }}\in {\mathcal {M}^{q}_\succcurlyeq [\mathbb {R};(s_j)_{j=0}^{2n},\preccurlyeq ]}$ and Remark 3.3, we see that $f\in {\mathcal {R}_{0,q}\left[ {\Pi _{\mathord {+}};(s_j)_{j=0}^{2n},\preccurlyeq } \right] }$. Hence, according to Theorem 6.8, regarding $(s_j)_{j=0}^{2n}\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,2n}$ and (13.1), we obtain $f\left( {w}\right) \in \mathscr {K}_{2n}\left( {w}\right)$. Because of $\sigma ={{\,\textrm{Rstr}\,}}_{\mathfrak {B}_{{[\alpha ,\infty )}}}{\hat{\sigma }}$, from Remark B.4(b) we get $\int _{{[\alpha ,\infty )}}\left( {x-w}\right) ^{-1}{\hat{\sigma }}\left( {\textrm{d}x}\right) =\int _{{[\alpha ,\infty )}}\left( {x-w}\right) ^{-1}\sigma \left( {\textrm{d}x}\right)$. Using Remark B.2, then $\int _\mathbb {R}\left( {x-w}\right) ^{-1}{\hat{\sigma }}\left( {\textrm{d}x}\right) =\int _{\mathbb {R}\backslash \left( {\mathbb {R}\backslash {[\alpha ,\infty )}}\right) }\left( {x-w}\right) ^{-1}{\hat{\sigma }}\left( {\textrm{d}x}\right) =\int _{{[\alpha ,\infty )}}\left( {x-w}\right) ^{-1}\sigma \left( {\textrm{d}x}\right)$ follows. Taking additionally into account (7.5), we conclude $F\left( {w}\right) =\int _{{[\alpha ,\infty )}}\left( {x-w}\right) ^{-1}\sigma \left( {\textrm{d}x}\right) =\int _{\mathbb {R}}\left( {x-w}\right) ^{-1}{\hat{\sigma }}\left( {\textrm{d}x}\right) =f\left( {w}\right)$. In view of $f\left( {w}\right) \in \mathscr {K}_{2n}\left( {w}\right)$, the proof is complete. $\square$

Suppose $\kappa \ge 1$. Let $(s_j)_{j=0}^{\kappa }\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,\kappa ,\alpha }$. Because of Remark 7.6, then . Let $m\in \mathbb {Z}_{0,\kappa -1}$ and let $z\in \mathbb {C}\backslash \mathbb {R}$. In accordance with Notation 7.1, we denote in the sequel by $\mathscr {A}_{{\alpha ,m}}\left( {z}\right)$, $\mathscr {B}_{{\alpha ,m}}\left( {z}\right)$, and $\mathscr {C}_{\alpha ,m}(z)$ the matrices given by Notation 6.6 for the sequence . Furthermore, let

$$\begin{aligned}{} & {} \mathscr {K}_{{{\mathord {\circ }},m}}\left( {z}\right) :=\Bigl \{\left( {z-\alpha }\right) ^{-1}\left( {X_{\mathord {\circ }}-s_{0}}\right) :\nonumber \\{} & {} \qquad \qquad X_{\mathord {\circ }}\in \mathfrak {K}\left( {\mathscr {C}_{{\alpha ,m}}\left( {z}\right) ;\left( {z-\overline{z}}\right) ^{-1}\mathscr {A}_{{\alpha ,m}}\left( {z}\right) ,\mathscr {B}_{{\alpha ,m}}\left( {z}\right) }\right) \Bigr \}. \end{aligned}$$

(13.3)

Lemma 13.2

Let $n\in \mathbb {N}_0$, let $(s_j)_{j=0}^{2n+1}\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,2n+1,\alpha }$, let $F\in {\mathcal {S}_{0,q}\left[ {\mathbb {C}\backslash {[\alpha ,\infty )};(s_j)_{j=0}^{2n+1},\preccurlyeq } \right] }$, and let $w\in \Pi _{\mathord {+}}$. Then $(a_j)_{j=0}^{2n}\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,2n}$ and $F\left( {w}\right) \in \mathscr {K}_{{{\mathord {\circ }},2n}}\left( {w}\right)$.

Proof

Lemma 7.5(b) yields $(a_j)_{j=0}^{2n}\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,2n}$, whereas Remark 7.8 shows $F\in \mathcal {S}_{0,q}\left( {\mathbb {C}\backslash {[\alpha ,\infty )}}\right)$ and that the ${[\alpha ,\infty )}$-spectral measure $\sigma$ of F belongs to ${\mathcal {M}^{q}_\succcurlyeq [{[\alpha ,\infty )};(s_j)_{j=0}^{2n+1},\preccurlyeq ]}$. In particular, (7.5) is valid as well as $\sigma \in \mathcal {M}_{\succcurlyeq ,2n+1}^{q}({[\alpha ,\infty )})$, (13.2) for all $j\in \mathbb {Z}_{0,2n}$, and $\int _{{[\alpha ,\infty )}}x^{2n+1}\sigma \left( {\textrm{d}x}\right) \preccurlyeq s_{2n+1}$ hold true. The function $c:{[\alpha ,\infty )}\rightarrow \mathbb {C}$ given by $c(x):=x-\alpha$ is continuous and, in particular, ${\mathfrak {B}_{{[\alpha ,\infty )}}}$-${\mathfrak {B}_{\mathbb {C}}}$-measurable. Regarding $\sigma \in \mathcal {M}_{\succcurlyeq ,2n+1}^{q}({[\alpha ,\infty )})$, we can infer $c\in \mathcal {L}^{1}\left( {{[\alpha ,\infty )},{\mathfrak {B}_{{[\alpha ,\infty )}}},\sigma ;\mathbb {C}}\right)$. Furthermore, we have $c\left( {{[\alpha ,\infty )}}\right) \subseteq [0,\infty )$. Thus, Proposition B.5(a) shows that $\rho :{\mathfrak {B}_{{[\alpha ,\infty )}}}\rightarrow \mathbb {C}^{{q\times q}}$ defined by $\rho (B):=\int _B\left( {x-\alpha }\right) \sigma \left( {\textrm{d}x}\right)$ belongs to ${\mathcal {M}_\succcurlyeq ^{q}({[\alpha ,\infty )})}$. Moreover, using $\sigma \in \mathcal {M}_{\succcurlyeq ,2n+1}^{q}({[\alpha ,\infty )})$ and Proposition B.5(b), it is easily checked that $\rho \in \mathcal {M}_{\succcurlyeq ,2n}^{q}({[\alpha ,\infty )})$ and that $\int _{{[\alpha ,\infty )}}x^{j+1}\sigma \left( {\textrm{d}x}\right) -\alpha \int _{{[\alpha ,\infty )}}x^j\sigma \left( {\textrm{d}x}\right) =\int _{{[\alpha ,\infty )}}x^j\left( {x-\alpha }\right) \sigma \left( {\textrm{d}x}\right) =\int _{{[\alpha ,\infty )}}x^j\rho \left( {\textrm{d}x}\right)$ holds true for all $j\in \mathbb {Z}_{0,2n}$. Taking additionally into account $\int _{{[\alpha ,\infty )}}x^{2n+1}\sigma \left( {\textrm{d}x}\right) \preccurlyeq s_{2n+1}$, (13.2), and (7.1), we then get $\int _{{[\alpha ,\infty )}}x^{2n}\rho \left( {\textrm{d}x}\right) \preccurlyeq s_{2n+1}-\alpha s_{2n}=a_{2n}$ and, in the case $n\ge 1$, for all $j\in \mathbb {Z}_{0,2n-1}$, moreover, $\int _{{[\alpha ,\infty )}}x^j\rho \left( {\textrm{d}x}\right) =s_{j+1}-\alpha s_{j}=a_{j}$. Consequently, $\rho \in {\mathcal {M}^{q}_\succcurlyeq [{[\alpha ,\infty )};(a_j)_{j=0}^{2n},\preccurlyeq ]}$. Regarding $\rho \in \mathcal {M}_{\succcurlyeq ,2n}^{q}({[\alpha ,\infty )})$, Remarks B.3 and B.4(b) show that ${\hat{\rho }}:{\mathfrak {B}_{\mathbb {R}}}\rightarrow \mathbb {C}^{{q\times q}}$ defined by ${\hat{\rho }}(B):=\rho \left( {B\cap {[\alpha ,\infty )}}\right)$ belongs to $\mathcal {M}_{\succcurlyeq ,2n}^{q}(\mathbb {R})$ and fulfills ${{\,\textrm{Rstr}\,}}_{\mathfrak {B}_{{[\alpha ,\infty )}}}{\hat{\rho }}=\rho$ and ${\hat{\rho }}\left( {\mathbb {R}\backslash {[\alpha ,\infty )}}\right) =O_{{q\times q}}$ as well as $\int _{{[\alpha ,\infty )}}x^j{\hat{\rho }}\left( {\textrm{d}x}\right) =\int _{{[\alpha ,\infty )}}x^j\rho \left( {\textrm{d}x}\right)$ for all $j\in \mathbb {Z}_{0,2n}$. Using Remark B.2, then $\int _\mathbb {R}x^j{\hat{\rho }}\left( {\textrm{d}x}\right) =\int _{\mathbb {R}\backslash \left( {\mathbb {R}\backslash {[\alpha ,\infty )}}\right) }x^j{\hat{\rho }}\left( {\textrm{d}x}\right) =\int _{{[\alpha ,\infty )}}x^j\rho \left( {\textrm{d}x}\right)$ for all $j\in \mathbb {Z}_{0,2n}$ follows. Hence, taking into account $\rho \in {\mathcal {M}^{q}_\succcurlyeq [{[\alpha ,\infty )};(a_j)_{j=0}^{2n},\preccurlyeq ]}$, we obtain ${\hat{\rho }}\in {\mathcal {M}^{q}_\succcurlyeq [\mathbb {R};(a_j)_{j=0}^{2n},\preccurlyeq ]}$. Taking Theorem 3.2 into account, we see that $g:\Pi _{\mathord {+}}\rightarrow \mathbb {C}^{{q\times q}}$ given by $g\left( {z}\right) :=\int _\mathbb {R}\left( {x-z}\right) ^{-1}{\hat{\rho }}\left( {\textrm{d}x}\right)$ is a well-defined matrix-valued function belonging to $\mathcal {R}_{0,q}\left( {\Pi _{\mathord {+}}}\right)$ and that ${\hat{\rho }}$ is the $\mathbb {R}$-spectral measure of g. Thus, in view of ${\hat{\rho }}\in {\mathcal {M}^{q}_\succcurlyeq [\mathbb {R};(a_j)_{j=0}^{2n},\preccurlyeq ]}$ and Remark 3.3, we see that $g\in {\mathcal {R}_{0,q}\left[ {\Pi _{\mathord {+}};(a_j)_{j=0}^{2n},\preccurlyeq } \right] }$. Since $(a_j)_{j=0}^{2n}$ belongs to $\mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,2n}$, using Theorem 6.8, we then obtain $g\left( {w}\right) \in \mathfrak {K}\left( {\mathscr {C}_{{\alpha ,2n}}\left( {w}\right) ;\left( {w-\overline{w}}\right) ^{-1}\mathscr {A}_{{\alpha ,2n}}\left( {w}\right) ,\mathscr {B}_{{\alpha ,2n}}\left( {w}\right) }\right)$. Because of $\rho ={{\,\textrm{Rstr}\,}}_{\mathfrak {B}_{{[\alpha ,\infty )}}}{\hat{\rho }}$, from Remark B.4(b) we get $\int _{{[\alpha ,\infty )}}\left( {x-w}\right) ^{-1}{\hat{\rho }}\left( {\textrm{d}x}\right) =\int _{{[\alpha ,\infty )}}\left( {x-w}\right) ^{-1}\rho \left( {\textrm{d}x}\right)$. Using Remark B.2, then $\int _\mathbb {R}\left( {x-w}\right) ^{-1}{\hat{\rho }}\left( {\textrm{d}x}\right) =\int _{\mathbb {R}\backslash \left( {\mathbb {R}\backslash {[\alpha ,\infty )}}\right) }\left( {x-w}\right) ^{-1}{\hat{\rho }}\left( {\textrm{d}x}\right) =\int _{{[\alpha ,\infty )}}\left( {x-w}\right) ^{-1}\rho \left( {\textrm{d}x}\right)$ follows. Taking additionally into account (13.2), (7.5), and Proposition B.5(b) we conclude

$$\begin{aligned}\begin{aligned} s_{0}+\left( {w-\alpha }\right) F\left( {w}\right)&=\int _{{[\alpha ,\infty )}}x^0\sigma \left( {\textrm{d}x}\right) +\left( {w-\alpha }\right) \int _{{[\alpha ,\infty )}}\frac{1}{x-w}\sigma \left( {\textrm{d}x}\right) \\&=\int _{{[\alpha ,\infty )}}\frac{x-w}{x-w}\sigma \left( {\textrm{d}x}\right) +\left( {w-\alpha }\right) \int _{{[\alpha ,\infty )}}\frac{1}{x-w}\sigma \left( {\textrm{d}x}\right) \\&=\int _{{[\alpha ,\infty )}}\frac{x-w+w-\alpha }{x-w}\sigma \left( {\textrm{d}x}\right) =\int _{{[\alpha ,\infty )}}\frac{x-\alpha }{x-w}\sigma \left( {\textrm{d}x}\right) \\&=\int _{{[\alpha ,\infty )}}\frac{1}{x-w}\rho \left( {\textrm{d}x}\right) =\int _\mathbb {R}\frac{1}{x-w}{\hat{\rho }}\left( {\textrm{d}x}\right) =g\left( {w}\right) \end{aligned}\end{aligned}$$

and, consequently, $F\left( {w}\right) =\left( {w-\alpha }\right) ^{-1}\left[ {g\left( {w}\right) -s_{0}} \right]$. Thus, using additionally $g\left( {w}\right) \in \mathfrak {K}\left( {\mathscr {C}_{{\alpha ,2n}}\left( {w}\right) ;\left( {w-\overline{w}}\right) ^{-1}\mathscr {A}_{{\alpha ,2n}}\left( {w}\right) ,\mathscr {B}_{{\alpha ,2n}}\left( {w}\right) }\right)$ and (13.3), we get $F\left( {w}\right) \in \mathscr {K}_{{{\mathord {\circ }},2n}}\left( {w}\right)$. $\square$

Proposition 13.3

Let $n\in \mathbb {N}_0$, let $(s_j)_{j=0}^{2n+1}\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,2n+1,\alpha }$, let $F\in {\mathcal {S}_{0,q}\left[ {\mathbb {C}\backslash {[\alpha ,\infty )};(s_j)_{j=0}^{2n+1},\preccurlyeq } \right] }$, and let $w\in \Pi _{\mathord {+}}$. Then the sequences $(s_j)_{j=0}^{2n}$ and $(a_j)_{j=0}^{2n}$ both belong to $\mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,2n}$ and $F\left( {w}\right) \in \mathscr {K}_{2n}\left( {w}\right) \cap \mathscr {K}_{{{\mathord {\circ }},2n}}\left( {w}\right)$ is fulfilled.

Proof

From Remark 7.3 we can infer $(s_j)_{j=0}^{2n}\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,2n,\alpha }$. Remark 7.8 provides $F\in \mathcal {S}_{0,q}\left( {\mathbb {C}\backslash {[\alpha ,\infty )}}\right)$ and that the ${[\alpha ,\infty )}$-spectral measure $\sigma$ of F belongs to ${\mathcal {M}^{q}_\succcurlyeq [{[\alpha ,\infty )};(s_j)_{j=0}^{2n+1},\preccurlyeq ]}$. In particular, then $\sigma \in {\mathcal {M}^{q}_\succcurlyeq [{[\alpha ,\infty )};(s_j)_{j=0}^{2n},\preccurlyeq ]}$. Regarding Remark 7.8, hence $F\in {\mathcal {S}_{0,q}\left[ {\mathbb {C}\backslash {[\alpha ,\infty )};(s_j)_{j=0}^{2n},\preccurlyeq } \right] }$. Thus, we can apply Lemma 13.1 to obtain $(s_j)_{j=0}^{2n}\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,2n}$ and $F\left( {w}\right) \in \mathscr {K}_{2n}\left( {w}\right)$. Furthermore, Lemma 13.2 immediately yields $(a_j)_{j=0}^{2n}\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,2n}$ and $F\left( {w}\right) \in \mathscr {K}_{{{\mathord {\circ }},2n}}\left( {w}\right)$. $\square$

Proposition 13.4

Let $n\in \mathbb {N}_0$, let $(s_j)_{j=0}^{2n+2}\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,2n+2,\alpha }$, let $F\in {\mathcal {S}_{0,q}\left[ {\mathbb {C}\backslash {[\alpha ,\infty )};(s_j)_{j=0}^{2n+2},\preccurlyeq } \right] }$, and let $w\in \Pi _{\mathord {+}}$. Then $(s_j)_{j=0}^{2n+2}\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,2n+2}$ as well as $(a_j)_{j=0}^{2n}\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,2n}$ and $F\left( {w}\right)$ belongs to $\mathscr {K}_{2n+2}\left( {w}\right) \cap \mathscr {K}_{{{\mathord {\circ }},2n}}\left( {w}\right)$.

Proof

Lemma 13.1 immediately yields $(s_j)_{j=0}^{2n+2}\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,2n+2}$ and $F\left( {w}\right) \in \mathscr {K}_{2n+2}\left( {w}\right)$. Furthermore, from Remark 7.3 we can infer $(s_j)_{j=0}^{2n+1}\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,2n+1,\alpha }$. Remark 7.8 provides $F\in \mathcal {S}_{0,q}\left( {\mathbb {C}\backslash {[\alpha ,\infty )}}\right)$ and that the ${[\alpha ,\infty )}$-spectral measure $\sigma$ of F belongs to ${\mathcal {M}^{q}_\succcurlyeq [{[\alpha ,\infty )};(s_j)_{j=0}^{2n+2},\preccurlyeq ]}$. In particular, then $\sigma \in {\mathcal {M}^{q}_\succcurlyeq [{[\alpha ,\infty )};(s_j)_{j=0}^{2n+1},\preccurlyeq ]}$. Regarding Remark 7.8, hence $F\in {\mathcal {S}_{0,q}\left[ {\mathbb {C}\backslash {[\alpha ,\infty )};(s_j)_{j=0}^{2n+1},\preccurlyeq } \right] }$. Thus, we can apply Lemma 13.2 to obtain $(a_j)_{j=0}^{2n}\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,2n}$ and $F\left( {w}\right) \in \mathscr {K}_{{{\mathord {\circ }},2n}}\left( {w}\right)$. $\square$

14 The Case of an Odd Number of Prescribed Matrix Moments

Now the set of values of Stieltjes transforms of all solutions of the truncated matricial Stieltjes moment problem under consideration is described. In this section, we focus on the case that an odd number m of moments is prescribed where $m\ge 3$. The case $m=1$ plays a certain special role and is studied in Sect. 16. We point out that in Lemmata 14.4 and 14.5 as well as in Lemmata 15.1 and 15.2 a coupling of a parametrization of the solution set of the truncated matricial Stieltjes moment problem with parametrizations of the solution sets of the two associated truncated matricial Hamburger moment problems is established.

Notation 14.1

Suppose $\kappa \ge 1$. Let $(s_j)_{j=0}^{\kappa }$ be a sequence of complex ${q\times q}$ matrices. Then, for each $m\in \mathbb {Z}_{1,\kappa }$, let $\textrm{V}^{\left( {\alpha }\right) }_{m}:\mathbb {C}\rightarrow \mathbb {C}^{{2q\times 2q}}$ be defined by

where $\mathfrak {k}_{-1}:=O_{{q\times q}}$. Moreover, let $\mathfrak {V}^{\left( {\alpha }\right) }_{2n+1}:=\textrm{V}^{\left( {\alpha }\right) }_{1}\textrm{V}^{\left( {\alpha }\right) }_{3}\cdots \textrm{V}^{\left( {\alpha }\right) }_{2n+1}$ for each $n\in \mathbb {N}_0$ such that $2n+1\le \kappa$, and, in the case $\kappa \ge 2$, let $\mathfrak {V}^{\left( {\alpha }\right) }_{2n+2}:=\textrm{V}^{\left( {\alpha }\right) }_{2}\textrm{V}^{\left( {\alpha }\right) }_{4}\cdots \textrm{V}^{\left( {\alpha }\right) }_{2n+2}$ for each $n\in \mathbb {N}_0$ fulfilling $2n+2\le \kappa$.

The combination of Lemma 14.2(a) below with Corollary 11.14 and (9.1) shows that the matrix polynomial $\mathfrak {V}^{\left( {\alpha }\right) }_{2n+1}$ generates the same linear fractional transformation as the matrix polynomial $\left[ \begin{array}{cc}-\mathfrak {a}_{n}\mathfrak {h}_{2n}^{\mathord {+}}&{}-\mathfrak {a}_{n+1}\\ \mathfrak {b}_{n}\mathfrak {h}_{2n}^{\mathord {+}}&{}\mathfrak {b}_{n+1}\end{array}\right]$ occurring in [6, Prop. 6.24] in the context of the truncated matricial Hamburger moment problem. Similarly, from Lemma 14.2(b) below, Corollary 12.17, and (9.2) it is seen that the matrix polynomial $\mathfrak {V}^{\left( {\alpha }\right) }_{2n+2}$ generates the same linear fractional transformation as the matrix polynomial $\left[ \begin{array}{cc}-\mathfrak {a}_{{\alpha ,n}}\mathfrak {h}_{{\alpha ,2n}}^{\mathord {+}}&{}-\mathfrak {a}_{{\alpha ,n+1}}\\ \mathfrak {b}_{{\alpha ,n}}\mathfrak {h}_{{\alpha ,2n}}^{\mathord {+}}&{}\mathfrak {b}_{{\alpha ,n+1}}\end{array}\right]$ associated to the sequence $(a_j)_{j=0}^{\kappa -1}$.

Lemma 14.2

Suppose $\kappa \ge 1$. Let $(s_j)_{j=0}^{\kappa }\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,\kappa ,\alpha }$ and let $z\in \mathbb {C}$.

• (a) For each $n\in \mathbb {N}_0$ fulfilling $2n+1\le \kappa$, then

  $$\begin{aligned} \mathfrak {V}^{\left( {\alpha }\right) }_{2n+1}\left( {z}\right) = \begin{bmatrix} -\textbf{a}_{n}\left( {z}\right) \mathfrak {k}_{2n}^{\mathord {+}}&{}-\textbf{a}_{n+1}\left( {z}\right) \\ \textbf{b}_{n}\left( {z}\right) \mathfrak {k}_{2n}^{\mathord {+}}&{}\textbf{b}_{n+1}\left( {z}\right) \end{bmatrix}. \end{aligned}$$

  (14.1)

• (b) If $\kappa \ge 2$, for each $n\in \mathbb {N}_0$ fulfilling $2n+2\le \kappa$, then

  $$\begin{aligned} \mathfrak {V}^{\left( {\alpha }\right) }_{2n+2}\left( {z}\right) = \begin{bmatrix} -\textbf{a}_{{{\mathord {\circ }},n}}\left( {z}\right) \mathfrak {k}_{2n+1}^{\mathord {+}}&{}-\textbf{a}_{{{\mathord {\circ }},n+1}}\left( {z}\right) \\ \textbf{b}_{{{\mathord {\circ }},n}}\left( {z}\right) \mathfrak {k}_{2n+1}^{\mathord {+}}&{}\textbf{b}_{{{\mathord {\circ }},n+1}}\left( {z}\right) \end{bmatrix}. \end{aligned}$$

  (14.2)

Proof

Let $\zeta :=z-\alpha$.

(a) From Notation 14.1 and Lemmata 11.6 and 11.5 we get

Thus, (14.1) holds true for $n=0$. In particular, part (a) is proved in the case $\kappa =1$. Now assume $\kappa \ge 2$. Then, there exists an integer $k \in \mathbb {N}$ with $2k+1\le \kappa$ such that (14.1) is fulfilled for $n= k-1$. Taking additionally into account Notation 14.1 and Remark 11.7, then

Consequently, part (a) is proved by induction.

(b) Assume $\kappa \ge 2$. Using Notations 14.1 and 5.5 and Lemmata 12.9 and 12.8, we obtain

Consequently, (14.2) holds true for $n=0$. In particular, part (b) is proved in the case $\kappa \le 3$. Now assume $\kappa \ge 4$. Then, there exists an integer $k \in \mathbb {N}$ with $2k+2\le \kappa$ such that (14.2) is fulfilled for $n= k-1$. Taking additionally into account Notation 14.1 and Remark 12.10, then

Thus, part (b) is also proved by induction. $\square$

Notation 14.3

Let $(s_j)_{j=0}^{\kappa }$ be a sequence of complex ${q\times q}$ matrices. For all $m\in \mathbb {Z}_{0,\kappa }$, let

$$\begin{aligned} \mathring{\mathbb {V}}_{m} :=\begin{bmatrix} -\varepsilon _{m}\textbf{q}_{m}\mathfrak {k}_{m}^{\mathord {+}}&{}-\textbf{q}_{m+1}\\ \varepsilon _{m}\textbf{p}_{m}\mathfrak {k}_{m}^{\mathord {+}}&{}\textbf{p}_{m+1} \end{bmatrix}. \end{aligned}$$

According to Notation 10.6, in particular

$$\begin{aligned} \mathring{\mathbb {V}}_{2n}\left( {z}\right) = \begin{bmatrix} -\left( {z-\alpha }\right) \textbf{q}_{2n}\left( {z}\right) \mathfrak {k}_{2n}^{\mathord {+}}&{}-\textbf{q}_{2n+1}\left( {z}\right) \\ \left( {z-\alpha }\right) \textbf{p}_{2n}\left( {z}\right) \mathfrak {k}_{2n}^{\mathord {+}}&{}\textbf{p}_{2n+1}\left( {z}\right) \end{bmatrix} \end{aligned}$$

(14.3)

holds true for all $n\in \mathbb {N}_0$ fulfilling $2n\le \kappa$ and all $z\in \mathbb {C}$ and

$$\begin{aligned} \mathring{\mathbb {V}}_{2n+1}\left( {z}\right) = \begin{bmatrix} -\textbf{q}_{2n+1}\left( {z}\right) \mathfrak {k}_{2n+1}^{\mathord {+}}&{}-\textbf{q}_{2n+2}\left( {z}\right) \\ \textbf{p}_{2n+1}\left( {z}\right) \mathfrak {k}_{2n+1}^{\mathord {+}}&{}\textbf{p}_{2n+2}\left( {z}\right) \end{bmatrix}. \end{aligned}$$

(14.4)

is valid for all $n\in \mathbb {N}_0$ fulfilling $2n+1\le \kappa$ and all $z\in \mathbb {C}$. Regarding Definition 9.1, this shows that the matrix polynomial $\mathring{\mathbb {V}}_{m}$ generates the linear fractional transformations given in Theorems 10.17 and 10.18 for parametrizing the solution set of Problem ${\textsf{IP}[\mathbb {C}\backslash {[\alpha ,\infty )};(s_j)_{j=0}^{m},\preccurlyeq ]}$.

Lemma 14.4

Suppose $\kappa \ge 2$. Let $(s_j)_{j=0}^{\kappa }\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,\kappa ,\alpha }$, let $n\in \mathbb {N}$ be such that $2n\le \kappa$, and let $z\in \mathbb {C}$. Then

$$\begin{aligned} \left( {z-\alpha }\right) \mathfrak {V}^{\left( {\alpha }\right) }_{2n}\left( {z}\right) \begin{bmatrix} \mathfrak {k}_{2n}\mathfrak {k}_{2n}^{\mathord {+}}&{}O_{{q\times q}}\\ \mathfrak {k}_{2n}^{\mathord {+}}&{}I_{q}\end{bmatrix} = \begin{bmatrix} \left( {z-\alpha }\right) I_{q}&{}s_{0}\\ O_{{q\times q}}&{}I_{q}\end{bmatrix}\mathring{\mathbb {V}}_{2n}\left( {z}\right) . \end{aligned}$$

Proof

Let $\zeta :=z-\alpha$. Using Lemma 14.2(b), Remarks 12.7 and 12.6, and Lemma 12.3, we obtain

(14.5)

Furthermore, (14.3), Lemma 11.3, (11.1), and (12.1) yield

(14.6)

Comparing (14.5) and (14.6) completes the proof. $\square$

Lemma 14.5

Suppose $\kappa \ge 1$. Let $(s_j)_{j=0}^{\kappa }\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,\kappa ,\alpha }$, let $n\in \mathbb {N}_0$ be such that $2n+1\le \kappa$, and let $z\in \mathbb {C}$. Then

$$\begin{aligned} \mathfrak {V}^{\left( {\alpha }\right) }_{2n+1}\left( {z}\right) \begin{bmatrix} \left( {z-\alpha }\right) I_{q}&{}\mathfrak {k}_{2n+1}\\ O_{{q\times q}}&{}I_{q}\end{bmatrix} = \mathring{\mathbb {V}}_{2n}\left( {z}\right) . \end{aligned}$$

Proof

Using Lemma 14.2(a), Lemma 11.3, and (11.1), we obtain

$$\begin{aligned} \mathfrak {V}^{\left( {\alpha }\right) }_{2n+1}\left( {z}\right) = \begin{bmatrix} -\textbf{a}_{n}\left( {z}\right) \mathfrak {k}_{2n}^{\mathord {+}}&{}-\textbf{a}_{n+1}\left( {z}\right) \\ \textbf{b}_{n}\left( {z}\right) \mathfrak {k}_{2n}^{\mathord {+}}&{}\textbf{b}_{n+1}\left( {z}\right) \end{bmatrix} = \begin{bmatrix} -\textbf{q}_{2n}\left( {z}\right) \mathfrak {k}_{2n}^{\mathord {+}}&{}-\textbf{q}_{2n+2}\left( {z}\right) \\ \textbf{p}_{2n}\left( {z}\right) \mathfrak {k}_{2n}^{\mathord {+}}&{}\textbf{p}_{2n+2}\left( {z}\right) \end{bmatrix}. \end{aligned}$$

Setting $\zeta :=z-\alpha$ and taking additionally into account (10.4), Notation 10.6, and (14.3), we can conclude then

$\square$

Notation 14.6

Suppose $\kappa \ge 1$. Let $(s_j)_{j=0}^{\kappa }$ be a sequence of complex ${q\times q}$ matrices. Then, let $\textrm{W}^{\left( {\alpha }\right) }_{1}:\mathbb {C}\rightarrow \mathbb {C}^{{2q\times 2q}}$ be defined by

If $\kappa \ge 2$, then, for each $m\in \mathbb {Z}_{2,\kappa }$, let $\textrm{W}^{\left( {\alpha }\right) }_{m}:\mathbb {C}\rightarrow \mathbb {C}^{{2q\times 2q}}$ be defined by

Moreover, let $\mathfrak {W}^{\left( {\alpha }\right) }_{2n+1}:=\textrm{W}^{\left( {\alpha }\right) }_{2n+1}\textrm{W}^{\left( {\alpha }\right) }_{2n-1}\cdots \textrm{W}^{\left( {\alpha }\right) }_{1}$ for each $n\in \mathbb {N}_0$ fulfilling $2n+1\le \kappa$ and, in the case $\kappa \ge 2$, let $\mathfrak {W}^{\left( {\alpha }\right) }_{2n+2}:=\textrm{W}^{\left( {\alpha }\right) }_{2n+2}\textrm{W}^{\left( {\alpha }\right) }_{2n}\cdots \textrm{W}^{\left( {\alpha }\right) }_{2}$ for each $n\in \mathbb {N}_0$ fulfilling $2n+2\le \kappa$.

Lemma 14.7

Suppose $\kappa \ge 2$. Let $(s_j)_{j=0}^{\kappa }\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,\kappa ,\alpha }$. For each $n\in \mathbb {N}_0$ fulfilling $2n+2\le \kappa$, let $\textbf{T}^{\left( {\alpha }\right) }_{2n+2}:\mathbb {C}\rightarrow \mathbb {C}^{{q\times q}}$ be defined by

$$\begin{aligned} \textbf{T}^{\left( {\alpha }\right) }_{2n+2}\left( {z}\right) :={\left\{ \begin{array}{ll} O_{{q\times q}},&{}\text { if }n=0\\ {\sum }_{\ell =0}^{n-1}\left( {z-\alpha }\right) ^{n-\ell }\left( {\mathfrak {k}_{2\ell +1}^{\mathord {+}}\mathfrak {k}_{2\ell +1}-\mathfrak {k}_{2\ell +2}^{\mathord {+}}\mathfrak {k}_{2\ell +2}}\right) ,&{}\text { if }n\ge 1 \end{array}\right. }.\nonumber \\ \end{aligned}$$

(14.7)

For each $n\in \mathbb {N}_0$ fulfilling $2n+2\le \kappa$ and all $z\in \mathbb {C}$, then

$$\begin{aligned}{} & {} \mathfrak {W}^{\left( {\alpha }\right) }_{2n+2}\left( {z}\right) \mathfrak {V}^{\left( {\alpha }\right) }_{2n+2}\left( {z}\right) \nonumber \\{} & {} \quad ={{\,\textrm{diag}\,}}\left( {\mathfrak {k}_{2n+1}\mathfrak {k}_{2n+1}^{\mathord {+}},\left( {z-\alpha }\right) ^{n+1}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) +\mathfrak {k}_{2n+1}^{\mathord {+}}\mathfrak {k}_{2n+1}+\textbf{T}^{\left( {\alpha }\right) }_{2n+2}\left( {z}\right) }\right) .\nonumber \\ \end{aligned}$$

(14.8)

Proof

Let $z\in \mathbb {C}$ and let $\zeta :=z-\alpha$. First note that Remark 9.3 yields (9.3) and (9.4). Our proof uses mathematical induction and is divided into three parts.

Part 1: Because of Notations 14.6 and 14.1, (9.3), and (9.4), we have

(14.9)

Regarding (14.7), hence (14.8) is checked for $n=0$. In particular, the proof is complete in the case $\kappa \le 3$.

Part 2: Assume $\kappa \ge 4$. In view of Notation 14.1, (14.9), (9.3), and (9.4), we see that

(14.10)

holds true. Due to (9.4), yielding

$$\begin{aligned}\begin{aligned}&\left[ {\zeta \left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) +\mathfrak {k}_{1}^{\mathord {+}}\mathfrak {k}_{1}} \right] \left[ {\zeta I_{q}-\left( {\mathfrak {k}_{2}^{\mathord {+}}\mathfrak {k}_{3}+\mathfrak {k}_{3}^{\mathord {+}}\mathfrak {k}_{4}}\right) } \right] \\&=\zeta ^2\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) -\zeta \left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) \left( {\mathfrak {k}_{2}^{\mathord {+}}\mathfrak {k}_{3}+\mathfrak {k}_{3}^{\mathord {+}}\mathfrak {k}_{4}}\right) +\zeta \mathfrak {k}_{1}^{\mathord {+}}\mathfrak {k}_{1}-\mathfrak {k}_{1}^{\mathord {+}}\mathfrak {k}_{1}\left( {\mathfrak {k}_{2}^{\mathord {+}}\mathfrak {k}_{3}+\mathfrak {k}_{3}^{\mathord {+}}\mathfrak {k}_{4}}\right) \\&=\zeta ^2\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) +\zeta \mathfrak {k}_{1}^{\mathord {+}}\mathfrak {k}_{1}-\left( {\mathfrak {k}_{2}^{\mathord {+}}\mathfrak {k}_{3}+\mathfrak {k}_{3}^{\mathord {+}}\mathfrak {k}_{4}}\right) , \end{aligned}\end{aligned}$$

equation (14.10) can be further simplified to

Using additionally Notation 14.6, (9.3), (9.4), and (14.7) we consequently obtain

(14.11)

Hence, (14.8) is checked for $n=1$. In particular, the proof is complete in the case $\kappa \le 5$.

Part 3: Now assume $\kappa \ge 6$. Then, in view of (14.11), there exists an integer $k \in \mathbb {N}$ with $2\left( { k +1}\right) +2\le \kappa$ such that (14.8) is fulfilled for $n= k$. Taking additionally into account Notation 14.1 and (9.3), then

(14.12)

follows, where

$$\begin{aligned} L :=\left[ {\zeta ^{ k +1}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) +\mathfrak {k}_{2k+1}^{\mathord {+}}\mathfrak {k}_{2k+1}+\textbf{T}^{\left( {\alpha }\right) }_{2 k +2}\left( {z}\right) } \right] \mathfrak {k}_{2 k +3}^{\mathord {+}}\end{aligned}$$

(14.13)

and

$$\begin{aligned}{} & {} R :=\left[ {\zeta ^{ k +1}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) +\mathfrak {k}_{2k+1}^{\mathord {+}}\mathfrak {k}_{2k+1}+\textbf{T}^{\left( {\alpha }\right) }_{2 k +2}\left( {z}\right) } \right] \nonumber \\{} & {} \quad \times \left( {\zeta I_{q}-\mathfrak {k}_{2 k +2}^{\mathord {+}}\mathfrak {k}_{2 k +3}-\mathfrak {k}_{2 k +3}^{\mathord {+}}\mathfrak {k}_{2 k +4}}\right) . \end{aligned}$$

(14.14)

Due to (14.13), (9.4), and (14.7), we have

$$\begin{aligned} \begin{aligned} L&=\zeta ^{ k +1}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) \mathfrak {k}_{2 k +3}^{\mathord {+}}+\mathfrak {k}_{2k+1}^{\mathord {+}}\mathfrak {k}_{2k+1}\mathfrak {k}_{2 k +3}^{\mathord {+}}+\textbf{T}^{\left( {\alpha }\right) }_{2 k +2}\left( {z}\right) \mathfrak {k}_{2 k +3}^{\mathord {+}}\\&=\mathfrak {k}_{2 k +3}^{\mathord {+}}+\sum _{\ell =0}^{k-1}\zeta ^{k-\ell }\left( {\mathfrak {k}_{2\ell +1}^{\mathord {+}}\mathfrak {k}_{2\ell +1}-\mathfrak {k}_{2\ell +2}^{\mathord {+}}\mathfrak {k}_{2\ell +2}}\right) \mathfrak {k}_{2 k +3}^{\mathord {+}}=\mathfrak {k}_{2 k +3}^{\mathord {+}}, \end{aligned}\end{aligned}$$

(14.15)

whereas from (14.14), (9.4), and (14.7), we conclude

$$\begin{aligned} \begin{aligned}&R =\zeta ^{ k +2}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) +\zeta \mathfrak {k}_{2k+1}^{\mathord {+}}\mathfrak {k}_{2k+1}+\zeta \textbf{T}^{\left( {\alpha }\right) }_{2 k +2}\left( {z}\right) \\&\;-\zeta ^{ k +1}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) \mathfrak {k}_{2 k +2}^{\mathord {+}}\mathfrak {k}_{2 k +3}-\mathfrak {k}_{2k+1}^{\mathord {+}}\mathfrak {k}_{2k+1}\mathfrak {k}_{2 k +2}^{\mathord {+}}\mathfrak {k}_{2 k +3}-\textbf{T}^{\left( {\alpha }\right) }_{2 k +2}\left( {z}\right) \mathfrak {k}_{2 k +2}^{\mathord {+}}\mathfrak {k}_{2 k +3}\\&\;-\zeta ^{ k +1}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) \mathfrak {k}_{2 k +3}^{\mathord {+}}\mathfrak {k}_{2 k +4}-\mathfrak {k}_{2k+1}^{\mathord {+}}\mathfrak {k}_{2k+1}\mathfrak {k}_{2 k +3}^{\mathord {+}}\mathfrak {k}_{2 k +4}-\textbf{T}^{\left( {\alpha }\right) }_{2 k +2}\left( {z}\right) \mathfrak {k}_{2 k +3}^{\mathord {+}}\mathfrak {k}_{2 k +4}\\&=\zeta ^{ k +2}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) +\zeta \mathfrak {k}_{2k+1}^{\mathord {+}}\mathfrak {k}_{2k+1}+\zeta \textbf{T}^{\left( {\alpha }\right) }_{2 k +2}\left( {z}\right) \\&\qquad -\mathfrak {k}_{2 k +2}^{\mathord {+}}\mathfrak {k}_{2 k +3}-\textbf{T}^{\left( {\alpha }\right) }_{2 k +2}\left( {z}\right) \mathfrak {k}_{2 k +2}^{\mathord {+}}\mathfrak {k}_{2 k +3}-\mathfrak {k}_{2 k +3}^{\mathord {+}}\mathfrak {k}_{2 k +4}-\textbf{T}^{\left( {\alpha }\right) }_{2 k +2}\left( {z}\right) \mathfrak {k}_{2 k +3}^{\mathord {+}}\mathfrak {k}_{2 k +4}\\&=\zeta ^{ k +2}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) +\zeta \mathfrak {k}_{2k+1}^{\mathord {+}}\mathfrak {k}_{2k+1}+\zeta \sum _{\ell =0}^{k-1}\zeta ^{k-\ell }\left( {\mathfrak {k}_{2\ell +1}^{\mathord {+}}\mathfrak {k}_{2\ell +1}-\mathfrak {k}_{2\ell +2}^{\mathord {+}}\mathfrak {k}_{2\ell +2}}\right) \\&\qquad -\mathfrak {k}_{2 k +2}^{\mathord {+}}\mathfrak {k}_{2 k +3}-\sum _{\ell =0}^{k-1}\zeta ^{k-\ell }\left( {\mathfrak {k}_{2\ell +1}^{\mathord {+}}\mathfrak {k}_{2\ell +1}-\mathfrak {k}_{2\ell +2}^{\mathord {+}}\mathfrak {k}_{2\ell +2}}\right) \mathfrak {k}_{2 k +2}^{\mathord {+}}\mathfrak {k}_{2 k +3}\\&\qquad -\mathfrak {k}_{2 k +3}^{\mathord {+}}\mathfrak {k}_{2 k +4}-\sum _{\ell =0}^{k-1}\zeta ^{k-\ell }\left( {\mathfrak {k}_{2\ell +1}^{\mathord {+}}\mathfrak {k}_{2\ell +1}-\mathfrak {k}_{2\ell +2}^{\mathord {+}}\mathfrak {k}_{2\ell +2}}\right) \mathfrak {k}_{2 k +3}^{\mathord {+}}\mathfrak {k}_{2 k +4}\\&=\zeta ^{ k +2}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) +\zeta \mathfrak {k}_{2k+1}^{\mathord {+}}\mathfrak {k}_{2k+1}+\sum _{\ell =0}^{k-1}\zeta ^{k+1-\ell }\left( {\mathfrak {k}_{2\ell +1}^{\mathord {+}}\mathfrak {k}_{2\ell +1}-\mathfrak {k}_{2\ell +2}^{\mathord {+}}\mathfrak {k}_{2\ell +2}}\right) \\&\qquad -\mathfrak {k}_{2 k +2}^{\mathord {+}}\mathfrak {k}_{2 k +3}-\mathfrak {k}_{2 k +3}^{\mathord {+}}\mathfrak {k}_{2 k +4}. \end{aligned}\end{aligned}$$

(14.16)

From Notation 14.6, (14.12), and (14.15) we get

(14.17)

Let

$$\begin{aligned} \mathfrak {W}^{\left( {\alpha }\right) }_{2\left( { k +1}\right) +2}\left( {z}\right) \mathfrak {V}^{\left( {\alpha }\right) }_{2\left( { k +1}\right) +2}\left( {z}\right) = \begin{bmatrix} A &{} B \\ C &{} D \end{bmatrix} \end{aligned}$$

(14.18)

be the ${q\times q}$ block representation of $\mathfrak {W}^{\left( {\alpha }\right) }_{2\left( { k +1}\right) +2}\left( {z}\right) \mathfrak {V}^{\left( {\alpha }\right) }_{2\left( { k +1}\right) +2}\left( {z}\right)$. From (14.18) and (14.17) we obtain $A =\mathfrak {k}_{2k+3}\mathfrak {k}_{2k+3}^{\mathord {+}}$, whereas (14.18), (14.17), and (9.4) yield $C=\left( {I_{q}-\mathfrak {k}_{2 k +2}^{\mathord {+}}\mathfrak {k}_{2 k +2}}\right) \mathfrak {k}_{2 k +3}^{\mathord {+}}=O_{{q\times q}}$. Moreover, in view of (14.18), (14.17), (14.16), and (9.3), it follows

$$\begin{aligned} B= & {} -\left( {\zeta I_{q}-\mathfrak {k}_{2 k +3}\mathfrak {k}_{2 k +2}^{\mathord {+}}-\mathfrak {k}_{2 k +4}\mathfrak {k}_{2 k +3}^{\mathord {+}}}\right) \mathfrak {k}_{2 k +3}+\mathfrak {k}_{2 k +3}R\\= & {} -\zeta \mathfrak {k}_{2 k +3}+\mathfrak {k}_{2 k +3}\mathfrak {k}_{2 k +2}^{\mathord {+}}\mathfrak {k}_{2 k +3}+\mathfrak {k}_{2 k +4}\mathfrak {k}_{2 k +3}^{\mathord {+}}\mathfrak {k}_{2 k +3}\\{} & {} \qquad +\mathfrak {k}_{2 k +3}\biggl \{\zeta ^{ k +2}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) +\zeta \mathfrak {k}_{2k+1}^{\mathord {+}}\mathfrak {k}_{2k+1}\\{} & {} \qquad \quad +\sum _{\ell =0}^{k-1}\zeta ^{k+1-\ell }\left( {\mathfrak {k}_{2\ell +1}^{\mathord {+}}\mathfrak {k}_{2\ell +1}-\mathfrak {k}_{2\ell +2}^{\mathord {+}}\mathfrak {k}_{2\ell +2}}\right) -\mathfrak {k}_{2 k +2}^{\mathord {+}}\mathfrak {k}_{2 k +3}-\mathfrak {k}_{2 k +3}^{\mathord {+}}\mathfrak {k}_{2 k +4}\biggr \}\\= & {} -\zeta \mathfrak {k}_{2 k +3}+\mathfrak {k}_{2 k +3}\mathfrak {k}_{2 k +2}^{\mathord {+}}\mathfrak {k}_{2 k +3}+\mathfrak {k}_{2 k +4}\mathfrak {k}_{2 k +3}^{\mathord {+}}\mathfrak {k}_{2 k +3}\\{} & {} \qquad +\zeta ^{ k +2}\mathfrak {k}_{2 k +3}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) +\zeta \mathfrak {k}_{2 k +3}\mathfrak {k}_{2k+1}^{\mathord {+}}\mathfrak {k}_{2k+1}\\{} & {} \qquad +\sum _{\ell =0}^{k-1}\zeta ^{k+1-\ell }\mathfrak {k}_{2 k +3}\left( {\mathfrak {k}_{2\ell +1}^{\mathord {+}}\mathfrak {k}_{2\ell +1}-\mathfrak {k}_{2\ell +2}^{\mathord {+}}\mathfrak {k}_{2\ell +2}}\right) \\{} & {} \qquad -\mathfrak {k}_{2 k +3}\mathfrak {k}_{2 k +2}^{\mathord {+}}\mathfrak {k}_{2 k +3}-\mathfrak {k}_{2 k +3}\mathfrak {k}_{2 k +3}^{\mathord {+}}\mathfrak {k}_{2 k +4}\\= & {} -\zeta \mathfrak {k}_{2 k +3}+\mathfrak {k}_{2 k +3}\mathfrak {k}_{2 k +2}^{\mathord {+}}\mathfrak {k}_{2 k +3}+\mathfrak {k}_{2 k +4}+\zeta \mathfrak {k}_{2 k +3}-\mathfrak {k}_{2 k +3}\mathfrak {k}_{2 k +2}^{\mathord {+}}\mathfrak {k}_{2 k +3}-\mathfrak {k}_{2 k +4}\\= & {} O_{{q\times q}}. \end{aligned}$$

Lastly, regarding (14.18), (14.17), and (14.16) and taking into account (9.3), (9.4), and (14.7), we conclude

$$\begin{aligned} D= & {} \mathfrak {k}_{2 k +3}^{\mathord {+}}\mathfrak {k}_{2 k +3}+\left( {I_{q}-\mathfrak {k}_{2 k +2}^{\mathord {+}}\mathfrak {k}_{2 k +2}}\right) R =\mathfrak {k}_{2 k +3}^{\mathord {+}}\mathfrak {k}_{2 k +3}+R-\mathfrak {k}_{2 k +2}^{\mathord {+}}\mathfrak {k}_{2 k +2}R\\= & {} \mathfrak {k}_{2 k +3}^{\mathord {+}}\mathfrak {k}_{2 k +3}+R-\mathfrak {k}_{2 k +2}^{\mathord {+}}\mathfrak {k}_{2 k +2}\biggl \{\zeta ^{ k +2}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) +\zeta \mathfrak {k}_{2k+1}^{\mathord {+}}\mathfrak {k}_{2k+1}\\{} & {} \qquad +\sum _{\ell =0}^{k-1}\zeta ^{k+1-\ell }\left( {\mathfrak {k}_{2\ell +1}^{\mathord {+}}\mathfrak {k}_{2\ell +1}-\mathfrak {k}_{2\ell +2}^{\mathord {+}}\mathfrak {k}_{2\ell +2}}\right) -\mathfrak {k}_{2 k +2}^{\mathord {+}}\mathfrak {k}_{2 k +3}-\mathfrak {k}_{2 k +3}^{\mathord {+}}\mathfrak {k}_{2 k +4}\biggr \}\\= & {} \mathfrak {k}_{2 k +3}^{\mathord {+}}\mathfrak {k}_{2 k +3}+R-\zeta ^{ k +2}\mathfrak {k}_{2 k +2}^{\mathord {+}}\mathfrak {k}_{2 k +2}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) -\zeta \mathfrak {k}_{2 k +2}^{\mathord {+}}\mathfrak {k}_{2 k +2}\mathfrak {k}_{2k+1}^{\mathord {+}}\mathfrak {k}_{2k+1}\\{} & {} \qquad -\sum _{\ell =0}^{k-1}\zeta ^{k+1-\ell }\mathfrak {k}_{2 k +2}^{\mathord {+}}\mathfrak {k}_{2 k +2}\left( {\mathfrak {k}_{2\ell +1}^{\mathord {+}}\mathfrak {k}_{2\ell +1}-\mathfrak {k}_{2\ell +2}^{\mathord {+}}\mathfrak {k}_{2\ell +2}}\right) \\{} & {} \qquad +\mathfrak {k}_{2 k +2}^{\mathord {+}}\mathfrak {k}_{2 k +2}\mathfrak {k}_{2 k +2}^{\mathord {+}}\mathfrak {k}_{2 k +3}+\mathfrak {k}_{2 k +2}^{\mathord {+}}\mathfrak {k}_{2 k +2}\mathfrak {k}_{2 k +3}^{\mathord {+}}\mathfrak {k}_{2 k +4}\\= & {} \mathfrak {k}_{2 k +3}^{\mathord {+}}\mathfrak {k}_{2 k +3}+R-\zeta \mathfrak {k}_{2 k +2}^{\mathord {+}}\mathfrak {k}_{2 k +2}+\mathfrak {k}_{2 k +2}^{\mathord {+}}\mathfrak {k}_{2 k +3}+\mathfrak {k}_{2 k +3}^{\mathord {+}}\mathfrak {k}_{2 k +4}\\= & {} \mathfrak {k}_{2 k +3}^{\mathord {+}}\mathfrak {k}_{2 k +3}+\biggl \{\zeta ^{ k +2}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) +\zeta \mathfrak {k}_{2k+1}^{\mathord {+}}\mathfrak {k}_{2k+1}\\{} & {} \qquad +\sum _{\ell =0}^{k-1}\zeta ^{k+1-\ell }\left( {\mathfrak {k}_{2\ell +1}^{\mathord {+}}\mathfrak {k}_{2\ell +1}-\mathfrak {k}_{2\ell +2}^{\mathord {+}}\mathfrak {k}_{2\ell +2}}\right) -\mathfrak {k}_{2 k +2}^{\mathord {+}}\mathfrak {k}_{2 k +3}-\mathfrak {k}_{2 k +3}^{\mathord {+}}\mathfrak {k}_{2 k +4}\biggr \}\\{} & {} \qquad -\zeta \mathfrak {k}_{2 k +2}^{\mathord {+}}\mathfrak {k}_{2 k +2}+\mathfrak {k}_{2 k +2}^{\mathord {+}}\mathfrak {k}_{2 k +3}+\mathfrak {k}_{2 k +3}^{\mathord {+}}\mathfrak {k}_{2 k +4}\\= & {} \mathfrak {k}_{2 k +3}^{\mathord {+}}\mathfrak {k}_{2 k +3}+\zeta ^{ k +2}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) \\{} & {} \qquad +\zeta \left( {\mathfrak {k}_{2k+1}^{\mathord {+}}\mathfrak {k}_{2k+1}-\mathfrak {k}_{2 k +2}^{\mathord {+}}\mathfrak {k}_{2 k +2}}\right) +\sum _{\ell =0}^{k-1}\zeta ^{k+1-\ell }\left( {\mathfrak {k}_{2\ell +1}^{\mathord {+}}\mathfrak {k}_{2\ell +1}-\mathfrak {k}_{2\ell +2}^{\mathord {+}}\mathfrak {k}_{2\ell +2}}\right) \\= & {} \zeta ^{ k +2}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) +\mathfrak {k}_{2 k +3}^{\mathord {+}}\mathfrak {k}_{2 k +3}+\sum _{\ell =0}^{k}\zeta ^{\left( {k+1}\right) -\ell }\left( {\mathfrak {k}_{2\ell +1}^{\mathord {+}}\mathfrak {k}_{2\ell +1}-\mathfrak {k}_{2\ell +2}^{\mathord {+}}\mathfrak {k}_{2\ell +2}}\right) \\= & {} \zeta ^{k+2}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) +\mathfrak {k}_{2k+3}^{\mathord {+}}\mathfrak {k}_{2k+3}+\textbf{T}^{\left( {\alpha }\right) }_{2\left( {k+1}\right) +2}\left( {z}\right) . \end{aligned}$$

Hence, in view of (14.18), we have

$$\begin{aligned}{} & {} \mathfrak {W}^{\left( {\alpha }\right) }_{2\left( { k +1}\right) +2}\left( {z}\right) \mathfrak {V}^{\left( {\alpha }\right) }_{2\left( { k +1}\right) +2}\left( {z}\right) = \begin{bmatrix} A &{} B \\ C &{} D \end{bmatrix}\\{} & {} \quad ={{\,\textrm{diag}\,}}\left( {\mathfrak {k}_{2k+3}\mathfrak {k}_{2k+3}^{\mathord {+}},\zeta ^{k+2}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) +\mathfrak {k}_{2k+3}^{\mathord {+}}\mathfrak {k}_{2k+3}+\textbf{T}^{\left( {\alpha }\right) }_{2\left( {k+1}\right) +2}\left( {z}\right) }\right) . \end{aligned}$$

Thus, (14.8) is checked for $n=k+1$ as well. Consequently, the assertion is inductively proved. $\square$

In order to avoid a cumbersome technical argumentation in the proof of the following proposition, the sequence of prescribed matrix moments is first extended to a sequence $(s_j)_{j=0}^{\infty }\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,\infty ,\alpha }$ and the sequence $(a_j)_{j=0}^{\infty }$ build according to Notation 7.1 is also considered. We further point out the important role of the different systems of matrix polynomials: We first use the $\mathbb {R}$-QMP $\left[ {(\mathfrak {a}_{k})_{k=0}^{\infty },(\mathfrak {b}_{k})_{k=0}^{\infty },(\mathfrak {c}_{k})_{k=0}^{\infty },(\mathfrak {d}_{k})_{k=0}^{\infty }} \right]$ and the first ${[\alpha ,\infty )}$-QMP $\left[ {(\textbf{a}_{k})_{k=0}^{\infty },(\textbf{b}_{k})_{k=0}^{\infty },(\textbf{c}_{k})_{k=0}^{\infty },(\textbf{d}_{k})_{k=0}^{\infty }} \right]$ associated with $(s_j)_{j=0}^{\infty }$ and after that we argue with the help of the $\mathbb {R}$-QMP $\left[ {(\mathfrak {a}_{{\alpha ,k}})_{k=0}^{\infty },(\mathfrak {b}_{{\alpha ,k}})_{k=0}^{\infty },(\mathfrak {c}_{{\alpha ,k}})_{k=0}^{\infty },(\mathfrak {d}_{{\alpha ,k}})_{k=0}^{\infty }} \right]$ associated with $(a_j)_{j=0}^{\infty }$ and the second ${[\alpha ,\infty )}$-QMP $\left[ {(\textbf{a}_{{{\mathord {\circ }},k}})_{k=0}^{\infty },(\textbf{b}_{{{\mathord {\circ }},k}})_{k=0}^{\infty },(\textbf{c}_{{{\mathord {\circ }},k}})_{k=0}^{\infty },(\textbf{d}_{{{\mathord {\circ }},k}})_{k=0}^{\infty }} \right]$ associated with $(s_j)_{j=0}^{\infty }$.

Proposition 14.8

Let $n\in \mathbb {N}_0$, let $(s_j)_{j=0}^{2n+2}\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,2n+2,\alpha }$, let $w\in \Pi _{\mathord {+}}$, and let $X\in \mathscr {K}_{2n+2}\left( {w}\right) \cap \mathscr {K}_{{{\mathord {\circ }},2n}}\left( {w}\right)$. Then there exists a function $F\in {\mathcal {S}_{0,q}\left[ {\mathbb {C}\backslash {[\alpha ,\infty )};(s_j)_{j=0}^{2n+2},\preccurlyeq } \right] }$ fulfilling $F\left( {w}\right) =X$.

Proof

According to Corollary 7.4 there exists a sequence $(s_{j})_{j=2n+3}^{\infty }$ of complex ${q\times q}$ matrices such that $(s_j)_{j=0}^{\infty }\in \mathcal {K}^\succcurlyeq _{q,\infty ,\alpha }=\mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,\infty ,\alpha }$. From [10, Prop. 5.1] we know that $(a_j)_{j=0}^{\infty }\in \mathcal {K}^\succcurlyeq _{q,\infty ,\alpha }=\mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,\infty ,\alpha }$. Consequently, from Remark 7.6 we can infer $(s_j)_{j=0}^{\infty }\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,\infty }=\mathcal {H}^\succcurlyeq _{q,\infty }$ and $(a_j)_{j=0}^{\infty }\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,\infty }=\mathcal {H}^\succcurlyeq _{q,\infty }$. Let $(\mathfrak {k}_{j})_{j=0}^{\infty }$ be the $\mathcal {K}_\alpha$-parameter sequence of $(s_j)_{j=0}^{\infty }$, let $(\mathfrak {h}_{j})_{j=0}^{\infty }$ be the $\mathcal {H}$-parameter sequence of $(s_j)_{j=0}^{\infty }$, and let $(\mathfrak {h}_{{\alpha ,j}})_{j=0}^{\infty }$ be the $\mathcal {H}$-parameter sequence of $(a_j)_{j=0}^{\infty }$. According to Remark 9.2, we have then

$$\begin{aligned} \mathfrak {k}_{2k}=\mathfrak {h}_{2k} \quad \text {and}\quad \mathfrak {k}_{2k+1}=\mathfrak {h}_{{\alpha ,2k}}\quad \text {for all }k\in \mathbb {N}_0. \end{aligned}$$

(14.19)

Remark 9.3 shows

$$\begin{aligned} \mathfrak {k}_{j}^*=\mathfrak {k}_{j}\quad \text {for all }j \in \mathbb {N}_0. \end{aligned}$$

(14.20)

Since $w\in \Pi _{\mathord {+}}$, we have $\Im w\in (0,\infty )$. Recall that $\mathbb {K}_{{q\times q}}$ stands for the set of all contractive complex ${q\times q}$ matrices. For the sake of improved readability, from hereon our proof is divided into six parts.

Part 1: Due to $X\in \mathscr {K}_{2n+2}\left( {w}\right) \cap \mathscr {K}_{{{\mathord {\circ }},2n}}\left( {w}\right)$, we have, in particular, $X\in \mathscr {K}_{2n+2}\left( {w}\right)$. Hence, according to (13.1) and Notation 6.7, there exists a matrix $C\in \mathbb {K}_{{q\times q}}$ fulfilling

$$\begin{aligned} X =\mathscr {C}_{2n+2}\left( {w}\right) +\left( {w-\overline{w}}\right) ^{-1}\mathscr {A}_{2n+2}\left( {w}\right) C\mathscr {B}_{2n+2}\left( {w}\right) . \end{aligned}$$

(14.21)

Regarding $(s_j)_{j=0}^{\infty }\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,\infty }$, let $(\chi _{j})_{j=-1}^{\infty }$ be the sequence of $\chi$-functions associated with $(s_j)_{j=0}^{\infty }$. Let $E:=-\left[ {\chi _{2n+3}\left( {w}\right) } \right] ^*$, let $B:=\left( {\Im w}\right) ^{-1}\Im E$, let $P:=\mathbb {P}_{\mathcal {R}\left( {E}\right) }$ and $Q:=\mathbb {P}_{\mathcal {N}\left( {E}\right) }$. Clearly, then

$$\begin{aligned} \chi _{2n+3}\left( {w}\right) =-E^*,\, \left[ {\chi _{2n+3}\left( {w}\right) } \right] ^*=-E,\, \,\text {and}\, \left( {\Im w}\right) ^{-1}\Im \chi _{2n+3}\left( {w}\right) =B.\nonumber \\ \end{aligned}$$

(14.22)

From Remark 6.3 we can thus infer $B\in \mathbb {C}_\succcurlyeq ^{{q\times q}}$. Obviously, the (constant) matrix-valued functions $\phi ,\psi :\Pi _{\mathord {+}}\rightarrow \mathbb {C}^{{q\times q}}$ defined by

$$\begin{aligned} \phi \left( {z}\right) :=E\sqrt{B}^{\mathord {+}}-E^*\sqrt{B}^{\mathord {+}}CP\text { and } \psi \left( {z}\right) :=\sqrt{B}^{\mathord {+}}-\sqrt{B}^{\mathord {+}}CP+Q,\qquad \end{aligned}$$

(14.23)

respectively, fulfill $\mathbb {H}\left( {\phi }\right) =\mathbb {H}\left( {\psi }\right) =\Pi _{\mathord {+}}$. Proposition 6.4 yields $\mathcal {R}\left( {\Im \chi _{2n+3}\left( {z}\right) }\right) =\mathcal {R}\left( {\left[ {\chi _{2n+3}\left( {z}\right) } \right] ^*}\right) =\mathcal {R}\left( {\mathfrak {h}_{2n+2}}\right)$ and $\mathcal {N}\left( {\Im \chi _{2n+3}\left( {z}\right) }\right) =\mathcal {N}\left( {\left[ {\chi _{2n+3}\left( {z}\right) } \right] ^*}\right) =\mathcal {N}\left( {\mathfrak {h}_{2n+2}}\right)$ for all $z\in \Pi _{\mathord {+}}$. By virtue of (14.22), consequently,

$$\begin{aligned} \mathcal {R}\left( {B}\right)&=\mathcal {R}\left( {E}\right) =\mathcal {R}\left( {\mathfrak {h}_{2n+2}}\right){} & {} \text {and}&\mathcal {N}\left( {B}\right)&=\mathcal {N}\left( {E}\right) =\mathcal {N}\left( {\mathfrak {h}_{2n+2}}\right) \end{aligned}$$

(14.24)

follow. Regarding $\Im w\in (0,\infty )$ and taking into account $B\in \mathbb {C}_\succcurlyeq ^{{q\times q}}$ and $\mathcal {R}\left( {B}\right) =\mathcal {R}\left( {E}\right)$ as well as (14.23) and $C\in \mathbb {K}_{{q\times q}}$, we can apply Lemma A.16 to discern that

$$\begin{aligned} {{\,\textrm{rank}\,}}\begin{bmatrix}\phi \left( {z}\right) \\ \psi \left( {z}\right) \end{bmatrix}=q \quad {and}\quad \Im \left( {\left[ {\psi \left( {z}\right) } \right] ^*\phi \left( {z}\right) }\right) \in \mathbb {C}_\succcurlyeq ^{{q\times q}}, \end{aligned}$$

(14.25)

that

$$\begin{aligned} P\phi \left( {z}\right) =\phi \left( {z}\right) ,\quad \phi \left( {z}\right) P=\phi \left( {z}\right) ,\quad \text {and}\qquad \psi \left( {z}\right) P=\psi \left( {z}\right) -Q, \end{aligned}$$

(14.26)

and that

$$\begin{aligned} \sqrt{B}^{\mathord {+}}\left[ {\phi \left( {z}\right) -E\psi \left( {z}\right) } \right] \left[ {\phi \left( {z}\right) -E^*\psi \left( {z}\right) } \right] ^{\mathord {+}}\sqrt{B} =PCP\end{aligned}$$

(14.27)

hold true for all $z\in \Pi _{\mathord {+}}$. In view of Definition 4.2 and Remark 4.3, from $\mathbb {H}\left( {\phi }\right) =\mathbb {H}\left( {\psi }\right) =\Pi _{\mathord {+}}$ and (14.25) we recognize that the pair $\left( {\phi };{\psi }\right)$ belongs to $\mathcal{P}\mathcal{R}_{q}\left( {\Pi _{\mathord {+}}}\right)$ and that the set $\mathcal {D}:=\emptyset$ belongs to $\mathscr {D}\left( {\phi ,\psi }\right)$. Regarding (14.24), we have

$$\begin{aligned} P&=\mathbb {P}_{\mathcal {R}\left( {E}\right) }=\mathbb {P}_{\mathcal {R}\left( {\mathfrak {h}_{2n+2}}\right) }{} & {} \text {and}&Q&=\mathbb {P}_{\mathcal {N}\left( {E}\right) }=\mathbb {P}_{\mathcal {N}\left( {\mathfrak {h}_{2n+2}}\right) }. \end{aligned}$$

(14.28)

From (14.28) and (14.26) we receive $\mathbb {P}_{\mathcal {R}\left( {\mathfrak {h}_{2n+2}}\right) }\phi \left( {z}\right) =\phi \left( {z}\right)$ for all $z\in \Pi _{\mathord {+}}$. According to Notation 4.5, hence $\left( {\phi };{\psi }\right) \in \mathcal {P}\left[ {\mathfrak {h}_{2n+2}} \right]$. Let $\left[ {(\mathfrak {a}_{k})_{k=0}^{\infty },(\mathfrak {b}_{k})_{k=0}^{\infty },(\mathfrak {c}_{k})_{k=0}^{\infty },(\mathfrak {d}_{k})_{k=0}^{\infty }} \right]$ be the $\mathbb {R}$-QMP associated with $(s_j)_{j=0}^{\infty }$ and let $\left[ {(\textbf{a}_{k})_{k=0}^{\infty },(\textbf{b}_{k})_{k=0}^{\infty },(\textbf{c}_{k})_{k=0}^{\infty },(\textbf{d}_{k})_{k=0}^{\infty }} \right]$ be the first ${[\alpha ,\infty )}$-QMP associated with $(s_j)_{j=0}^{\infty }$. Regarding $\mathcal {D}\in \mathscr {D}\left( {\phi ,\psi }\right)$ as well as $\mathcal {D}=\emptyset$, we can infer from Corollary 11.14 for all $z\in \Pi _{\mathord {+}}$ then

$$\begin{aligned} \det \left( {\mathfrak {b}_{n+1}\left( {z}\right) \mathfrak {h}_{2n+2}^{\mathord {+}}\phi \left( {z}\right) +\mathfrak {b}_{n+2}\left( {z}\right) \psi \left( {z}\right) }\right)&\ne 0, \end{aligned}$$

(14.29)

$$\begin{aligned} \det \left( {\textbf{b}_{n+1}\left( {z}\right) \mathfrak {h}_{2n+2}^{\mathord {+}}\phi \left( {z}\right) +\textbf{b}_{n+2}\left( {z}\right) \psi \left( {z}\right) }\right)&\ne 0, \end{aligned}$$

(14.30)

and

$$\begin{aligned}{} & {} -\left[ {\mathfrak {a}_{n+1}\left( {z}\right) \mathfrak {h}_{2n+2}^{\mathord {+}}\phi \left( {z}\right) +\mathfrak {a}_{n+2}\left( {z}\right) \psi \left( {z}\right) } \right] \left[ {\mathfrak {b}_{n+1}\left( {z}\right) \mathfrak {h}_{2n+2}^{\mathord {+}}\phi \left( {z}\right) +\mathfrak {b}_{n+2}\left( {z}\right) \psi \left( {z}\right) } \right] ^{-1}\nonumber \\{} & {} \quad =-\left[ {\textbf{a}_{n+1}\left( {z}\right) \mathfrak {h}_{2n+2}^{\mathord {+}}\phi \left( {z}\right) +\textbf{a}_{n+2}\left( {z}\right) \psi \left( {z}\right) } \right] \nonumber \\{} & {} \qquad \times \left[ {\textbf{b}_{n+1}\left( {z}\right) \mathfrak {h}_{2n+2}^{\mathord {+}}\phi \left( {z}\right) +\textbf{b}_{n+2}\left( {z}\right) \psi \left( {z}\right) } \right] ^{-1}. \nonumber \\ \end{aligned}$$

(14.31)

Setting

$$\begin{aligned} S&:=\phi \left( {w}\right){} & {} \text {and}&T&:=\psi \left( {w}\right) \end{aligned}$$

(14.32)

we see, in view of (14.29), (14.30), and (14.31), that

$$\begin{aligned} {\mathfrak {R}}&:=\mathfrak {b}_{n+1}\left( {w}\right) \mathfrak {h}_{2n+2}^{\mathord {+}}S+\mathfrak {b}_{n+2}\left( {w}\right) T{} & {} \text {and}&{\textbf{R}}&:=\textbf{b}_{n+1}\left( {w}\right) \mathfrak {h}_{2n+2}^{\mathord {+}}S+\textbf{b}_{n+2}\left( {w}\right) T\end{aligned}$$

(14.33)

satisfy $\det {\mathfrak {R}}\ne 0$ and $\det {\textbf{R}}\ne 0$ as well as

$$\begin{aligned}{} & {} -\left[ {\mathfrak {a}_{n+1}\left( {w}\right) \mathfrak {h}_{2n+2}^{\mathord {+}}S+\mathfrak {a}_{n+2}\left( {w}\right) T} \right] {\mathfrak {R}}^{-1}\nonumber \\{} & {} \quad =-\left[ {\textbf{a}_{n+1}\left( {w}\right) \mathfrak {h}_{2n+2}^{\mathord {+}}S+\textbf{a}_{n+2}\left( {w}\right) T} \right] {\textbf{R}}^{-1}. \end{aligned}$$

(14.34)

Now we are going to justify that all assumptions for the application of Proposition 6.11 to the sequence $(s_j)_{j=0}^{2n+3}$ are satisfied. Because of $(s_j)_{j=0}^{\infty }\in \mathcal {H}^\succcurlyeq _{q,\infty }$, we have $(s_j)_{j=0}^{2n+3}\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,2n+3}$. According to (14.28), we have $P=\mathbb {P}_{\mathcal {R}\left( {\mathfrak {h}_{2n+2}}\right) }$ and $Q=\mathbb {P}_{\mathcal {N}\left( {\mathfrak {h}_{2n+2}}\right) }$. In view of $\Im w\in (0,\infty )$ and (14.32), from (14.25) and (14.26), we see $\Im \left( {w}\right) \Im \left( {T^*S}\right) \in \mathbb {C}_\succcurlyeq ^{{q\times q}}$ as well as

$$\begin{aligned} PS=S,\quad SP=S,\quad \text {and}\quad TP=T-Q. \end{aligned}$$

(14.35)

Using additionally (14.33) and $\det {\mathfrak {R}}\ne 0$, we can thus apply Proposition 6.11 to infer that

$$\begin{aligned}{} & {} {\textbf{K}} :=\sqrt{\left( {\Im w}\right) ^{-1}\Im \chi _{2n+3}\left( {w}\right) }^{\mathord {+}}\left( {S+\left[ {\chi _{2n+3}\left( {w}\right) } \right] ^*T}\right) \nonumber \\{} & {} \quad \times \left( {S+\chi _{2n+3}\left( {w}\right) T}\right) ^{\mathord {+}}\sqrt{\left( {\Im w}\right) ^{-1}\Im \chi _{2n+3}\left( {w}\right) } \end{aligned}$$

(14.36)

is a contractive matrix which fulfills

$$\begin{aligned}{} & {} -\left[ {\mathfrak {a}_{n+1}\left( {w}\right) \mathfrak {h}_{2n+2}^{\mathord {+}}S+\mathfrak {a}_{n+2}\left( {w}\right) T} \right] {\mathfrak {R}}^{-1}\nonumber \\{} & {} \quad =\mathscr {C}_{2n+3}\left( {w}\right) +\left( {w-\overline{w}}\right) ^{-1}\mathscr {A}_{2n+3}\left( {w}\right) {\textbf{K}}\mathscr {B}_{2n+3}\left( {w}\right) . \end{aligned}$$

(14.37)

By virtue of (14.36), (14.22), (14.32), and (14.27), we discern

$$\begin{aligned} {\textbf{K}} =\sqrt{B}^{\mathord {+}}\left[ {\phi \left( {w}\right) -E\psi \left( {w}\right) } \right] \left[ {\phi \left( {w}\right) -E^*\psi \left( {w}\right) } \right] ^{\mathord {+}}\sqrt{B} =PCP. \end{aligned}$$

(14.38)

The application of Lemma 6.9 to the sequence $(s_j)_{j=0}^{\infty }\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,\infty }$ yields

$$\begin{aligned} \mathscr {A}_{2n+2}\left( {w}\right) =\mathscr {A}_{2n+3}\left( {w}\right) ,\; \mathscr {B}_{2n+2}\left( {w}\right) =\mathscr {B}_{2n+3}\left( {w}\right) , \text { and } \mathscr {C}_{2n+2}\left( {w}\right) =\mathscr {C}_{2n+3}\left( {w}\right) .\nonumber \\ \end{aligned}$$

(14.39)

From Proposition 6.10 we get $\mathcal {N}\left( {\mathscr {A}_{2n+2}\left( {w}\right) }\right) =\mathcal {N}\left( {\mathfrak {h}_{2n+2}}\right)$ and $\mathcal {R}\left( {\mathscr {B}_{2n+2}\left( {w}\right) }\right) =\mathcal {R}\left( {\mathfrak {h}_{2n+2}}\right)$. According to (14.28), then $P\mathscr {B}_{2n+2}\left( {w}\right) =\mathscr {B}_{2n+2}\left( {w}\right)$. By virtue of (14.28) and Remark A.8, we see furthermore $\mathcal {R}\left( {P}\right) =\mathcal {R}\left( {\mathfrak {h}_{2n+2}}\right)$ and $\mathcal {R}\left( {Q}\right) =\mathcal {N}\left( {\mathfrak {h}_{2n+2}}\right)$. Hence, we get $\mathscr {A}_{2n+2}\left( {w}\right) Q=O_{{q\times q}}$. Remark A.7 provides $\mathcal {N}\left( {\mathfrak {h}_{2n+2}^*}\right) =\mathcal {R}\left( {\mathfrak {h}_{2n+2}}\right) ^\bot$. Since, in view of $(s_j)_{j=0}^{\infty }\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,\infty }$, Remark 5.3 yields $\mathfrak {h}_{2n+2}^*=\mathfrak {h}_{2n+2}$, then $\mathcal {N}\left( {\mathfrak {h}_{2n+2}}\right) =\mathcal {R}\left( {\mathfrak {h}_{2n+2}}\right) ^\bot$ follows. Thus, Remark A.9 yields $P+Q=I_{q}$, implying $\mathscr {A}_{2n+2}\left( {w}\right) P=\mathscr {A}_{2n+2}\left( {w}\right)$. Consequently, we infer $\mathscr {A}_{2n+2}\left( {w}\right) PCP\mathscr {B}_{2n+2}\left( {w}\right) =\mathscr {A}_{2n+2}\left( {w}\right) C\mathscr {B}_{2n+2}\left( {w}\right)$. Using additionally (14.21), (14.39), (14.38), (14.37), and (14.34), we conclude then

$$\begin{aligned}\begin{aligned} X&=\mathscr {C}_{2n+2}\left( {w}\right) +\left( {w-\overline{w}}\right) ^{-1}\mathscr {A}_{2n+2}\left( {w}\right) C\mathscr {B}_{2n+2}\left( {w}\right) \\&=\mathscr {C}_{2n+2}\left( {w}\right) +\left( {w-\overline{w}}\right) ^{-1}\mathscr {A}_{2n+2}\left( {w}\right) PCP\mathscr {B}_{2n+2}\left( {w}\right) \\&=\mathscr {C}_{2n+3}\left( {w}\right) +\left( {w-\overline{w}}\right) ^{-1}\mathscr {A}_{2n+3}\left( {w}\right) {\textbf{K}}\mathscr {B}_{2n+3}\left( {w}\right) \\&=-\left[ {\mathfrak {a}_{n+1}\left( {w}\right) \mathfrak {h}_{2n+2}^{\mathord {+}}S+\mathfrak {a}_{n+2}\left( {w}\right) T} \right] {\mathfrak {R}}^{-1}\\&=-\left[ {\textbf{a}_{n+1}\left( {w}\right) \mathfrak {h}_{2n+2}^{\mathord {+}}S+\textbf{a}_{n+2}\left( {w}\right) T} \right] {\textbf{R}}^{-1}. \end{aligned}\end{aligned}$$

Using additionally $\det {\textbf{R}} \ne 0$, (14.33), (14.19), and Lemma 14.2(a), we then obtain

$$\begin{aligned} \begin{aligned}&\begin{bmatrix} X\\ I_{q}\end{bmatrix} =\begin{bmatrix} -\left[ {\textbf{a}_{n+1}\left( {w}\right) \mathfrak {h}_{2n+2}^{\mathord {+}}S+\textbf{a}_{n+2}\left( {w}\right) T} \right] {\textbf{R}}^{-1}\\ {\textbf{R}} {\textbf{R}}^{-1}\end{bmatrix}\\&=\begin{bmatrix} -\textbf{a}_{n+1}\left( {w}\right) \mathfrak {h}_{2n+2}^{\mathord {+}}S-\textbf{a}_{n+2}\left( {w}\right) T\\ \textbf{b}_{n+1}\left( {w}\right) \mathfrak {h}_{2n+2}^{\mathord {+}}S+\textbf{b}_{n+2}\left( {w}\right) T\end{bmatrix} {\textbf{R}}^{-1}\\&=\begin{bmatrix} -\textbf{a}_{n+1}\left( {w}\right) \mathfrak {k}_{2n+2}^{\mathord {+}}&{}-\textbf{a}_{n+2}\left( {w}\right) \\ \textbf{b}_{n+1}\left( {w}\right) \mathfrak {k}_{2n+2}^{\mathord {+}}&{}\textbf{b}_{n+2}\left( {w}\right) \end{bmatrix}\begin{bmatrix} S\\ T\end{bmatrix}{\textbf{R}}^{-1}=\mathfrak {V}^{\left( {\alpha }\right) }_{2n+3}\left( {w}\right) \begin{bmatrix} S\\ T\end{bmatrix}{\textbf{R}}^{-1}. \end{aligned}\end{aligned}$$

(14.40)

Part 2: Regarding the assumption $X\in \mathscr {K}_{2n+2}\left( {w}\right) \cap \mathscr {K}_{{{\mathord {\circ }},2n}}\left( {w}\right)$, in particular, $X\in \mathscr {K}_{{{\mathord {\circ }},2n}}\left( {w}\right)$. Then, according to (13.3) and Notation 6.7, there exists a matrix $C_{\mathord {\circ }}\in \mathbb {K}_{{q\times q}}$ such that

$$\begin{aligned} \eta X+s_{0} =\mathscr {C}_{{\alpha ,2n}}\left( {w}\right) +\left( {w-\overline{w}}\right) ^{-1}\mathscr {A}_{{\alpha ,2n}}\left( {w}\right) C_{\mathord {\circ }}\mathscr {B}_{{\alpha ,2n}}\left( {w}\right) , \end{aligned}$$

(14.41)

where $\eta :=w-\alpha$. Regarding $(a_j)_{j=0}^{\infty }\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,\infty }$, let $(\chi _{{\alpha ,j}})_{j=-1}^{\infty }$ be the sequence of $\chi$-functions associated with $(a_j)_{j=0}^{\infty }$. Let $E_{\mathord {\circ }}:=-\left[ {\chi _{{\alpha ,2n+1}}\left( {w}\right) } \right] ^*$, let $B_{\mathord {\circ }}:=\left( {\Im w}\right) ^{-1}\Im E_{\mathord {\circ }}$, let $P_{\mathord {\circ }}:=\mathbb {P}_{\mathcal {R}\left( {E_{\mathord {\circ }}}\right) }$ and $Q_{\mathord {\circ }}:=\mathbb {P}_{\mathcal {N}\left( {E_{\mathord {\circ }}}\right) }$. Clearly, then

$$\begin{aligned} \chi _{{\alpha ,2n+1}}\left( {w}\right) =-E_{\mathord {\circ }}^*,\; \left[ {\chi _{{\alpha ,2n+1}}\left( {w}\right) } \right] ^*=-E_{\mathord {\circ }}, \text { and } \left( {\Im w}\right) ^{-1}\Im \chi _{{\alpha ,2n+1}}\left( {w}\right) =B_{\mathord {\circ }}.\nonumber \\ \end{aligned}$$

(14.42)

From Remark 6.3 we can thus infer $B_{\mathord {\circ }}\in \mathbb {C}_\succcurlyeq ^{{q\times q}}$. Obviously, the (constant) matrix-valued functions $\phi _{\mathord {\circ }},\psi _{\mathord {\circ }}:\Pi _{\mathord {+}}\rightarrow \mathbb {C}^{{q\times q}}$ defined by

$$\begin{aligned} \phi _{\mathord {\circ }}\left( {z}\right) :=E_{\mathord {\circ }}\sqrt{B_{\mathord {\circ }}}^{\mathord {+}}-E_{\mathord {\circ }}^*\sqrt{B_{\mathord {\circ }}}^{\mathord {+}}C_{\mathord {\circ }}P_{\mathord {\circ }}\text { and } \psi _{\mathord {\circ }}\left( {z}\right) :=\sqrt{B_{\mathord {\circ }}}^{\mathord {+}}-\sqrt{B_{\mathord {\circ }}}^{\mathord {+}}C_{\mathord {\circ }}P_{\mathord {\circ }}+Q_{\mathord {\circ }}, \nonumber \\ \end{aligned}$$

(14.43)

respectively, fulfill $\mathbb {H}\left( {\phi _{\mathord {\circ }}}\right) =\mathbb {H}\left( {\psi _{\mathord {\circ }}}\right) =\Pi _{\mathord {+}}$. Proposition 6.4 yields $\mathcal {R}\left( {\Im \chi _{{\alpha ,2n+1}}\left( {z}\right) }\right) =\mathcal {R}\left( {\left[ {\chi _{{\alpha ,2n+1}}\left( {z}\right) } \right] ^*}\right) =\mathcal {R}\left( {\mathfrak {h}_{{\alpha ,2n}}}\right)$ and $\mathcal {N}\left( {\Im \chi _{{\alpha ,2n+1}}\left( {z}\right) }\right) =\mathcal {N}\left( {\left[ {\chi _{{\alpha ,2n+1}}\left( {z}\right) } \right] ^*}\right) =\mathcal {N}\left( {\mathfrak {h}_{{\alpha ,2n}}}\right)$ for all $z\in \Pi _{\mathord {+}}$. By virtue of (14.42), consequently,

$$\begin{aligned} \mathcal {R}\left( {B_{\mathord {\circ }}}\right)&=\mathcal {R}\left( {E_{\mathord {\circ }}}\right) =\mathcal {R}\left( {\mathfrak {h}_{{\alpha ,2n}}}\right){} & {} \text {and}&\mathcal {N}\left( {B_{\mathord {\circ }}}\right)&=\mathcal {N}\left( {E_{\mathord {\circ }}}\right) =\mathcal {N}\left( {\mathfrak {h}_{{\alpha ,2n}}}\right) \end{aligned}$$

(14.44)

follow. Regarding $\Im w\in (0,\infty )$ and taking into account $B_{\mathord {\circ }}\in \mathbb {C}_\succcurlyeq ^{{q\times q}}$ and $\mathcal {R}\left( {B_{\mathord {\circ }}}\right) =\mathcal {R}\left( {E_{\mathord {\circ }}}\right)$ as well as (14.43) and $C_{\mathord {\circ }}\in \mathbb {K}_{{q\times q}}$, we can apply Lemma A.16 to discern that

$$\begin{aligned} {{\,\textrm{rank}\,}}\begin{bmatrix}\phi _{\mathord {\circ }}\left( {z}\right) \\ \psi _{\mathord {\circ }}\left( {z}\right) \end{bmatrix}&=q{} & {} \text {and}&\Im \left( {\left[ {\psi _{\mathord {\circ }}\left( {z}\right) } \right] ^*\phi _{\mathord {\circ }}\left( {z}\right) }\right)&\in \mathbb {C}_\succcurlyeq ^{{q\times q}}, \end{aligned}$$

(14.45)

that

$$\begin{aligned} P_{\mathord {\circ }}\phi _{\mathord {\circ }}\left( {z}\right) =\phi _{\mathord {\circ }}\left( {z}\right) ,\; \phi _{\mathord {\circ }}\left( {z}\right) P_{\mathord {\circ }}=\phi _{\mathord {\circ }}\left( {z}\right) , \text { and } \psi _{\mathord {\circ }}\left( {z}\right) P_{\mathord {\circ }}=\psi _{\mathord {\circ }}\left( {z}\right) -Q_{\mathord {\circ }},\qquad \end{aligned}$$

(14.46)

and that

$$\begin{aligned} \sqrt{B_{\mathord {\circ }}}^{\mathord {+}}\left[ {\phi _{\mathord {\circ }}\left( {z}\right) -E_{\mathord {\circ }}\psi \left( {z}\right) } \right] \left[ {\phi _{\mathord {\circ }}\left( {z}\right) -E_{\mathord {\circ }}^*\psi _{\mathord {\circ }}\left( {z}\right) } \right] ^{\mathord {+}}\sqrt{B_{\mathord {\circ }}} =P_{\mathord {\circ }}C_{\mathord {\circ }}P_{\mathord {\circ }}\end{aligned}$$

(14.47)

hold true for all $z\in \Pi _{\mathord {+}}$. In view of Definition 4.2 and Remark 4.3, from $\mathbb {H}\left( {\phi _{\mathord {\circ }}}\right) =\mathbb {H}\left( {\psi _{\mathord {\circ }}}\right) =\Pi _{\mathord {+}}$ and (14.45) we recognize that the pair $\left( {\phi _{\mathord {\circ }}};{\psi _{\mathord {\circ }}}\right)$ belongs to $\mathcal{P}\mathcal{R}_{q}\left( {\Pi _{\mathord {+}}}\right)$ and that the set $\mathcal {D}_{\mathord {\circ }}:=\emptyset$ belongs to $\mathscr {D}\left( {\phi _{\mathord {\circ }},\psi _{\mathord {\circ }}}\right)$. Regarding (14.44), we have

$$\begin{aligned} P_{\mathord {\circ }}&=\mathbb {P}_{\mathcal {R}\left( {E_{\mathord {\circ }}}\right) }=\mathbb {P}_{\mathcal {R}\left( {\mathfrak {h}_{{\alpha ,2n}}}\right) }{} & {} \text {and}&Q_{\mathord {\circ }}&=\mathbb {P}_{\mathcal {N}\left( {E_{\mathord {\circ }}}\right) }=\mathbb {P}_{\mathcal {N}\left( {\mathfrak {h}_{{\alpha ,2n}}}\right) }. \end{aligned}$$

(14.48)

From (14.48) and (14.46) we receive $\mathbb {P}_{\mathcal {R}\left( {\mathfrak {h}_{{\alpha ,2n}}}\right) }\phi _{\mathord {\circ }}\left( {z}\right) =\phi _{\mathord {\circ }}\left( {z}\right)$ for all $z\in \Pi _{\mathord {+}}$. According to Notation 4.5, hence $\left( {\phi _{\mathord {\circ }}};{\psi _{\mathord {\circ }}}\right) \in \mathcal {P}\left[ {\mathfrak {h}_{{\alpha ,2n}}} \right]$. Let $\left[ {(\mathfrak {a}_{{\alpha ,k}})_{k=0}^{\infty },(\mathfrak {b}_{{\alpha ,k}})_{k=0}^{\infty },(\mathfrak {c}_{{\alpha ,k}})_{k=0}^{\infty },(\mathfrak {d}_{{\alpha ,k}})_{k=0}^{\infty }} \right]$ be the $\mathbb {R}$-QMP associated with $(a_j)_{j=0}^{\infty }$ and let $\left[ {(\textbf{a}_{{{\mathord {\circ }},k}})_{k=0}^{\infty },(\textbf{b}_{{{\mathord {\circ }},k}})_{k=0}^{\infty },(\textbf{c}_{{{\mathord {\circ }},k}})_{k=0}^{\infty },(\textbf{d}_{{{\mathord {\circ }},k}})_{k=0}^{\infty }} \right]$ be the second ${[\alpha ,\infty )}$-QMP associated with $(s_j)_{j=0}^{\infty }$. Regarding $\mathcal {D}_{\mathord {\circ }}\in \mathscr {D}\left( {\phi _{\mathord {\circ }},\psi _{\mathord {\circ }}}\right)$ as well as $\mathcal {D}_{\mathord {\circ }}=\emptyset$, we can infer from Corollary 12.17 for all $z\in \Pi _{\mathord {+}}$ then

$$\begin{aligned} \det \left( {\mathfrak {b}_{{\alpha ,n}}\left( {z}\right) \mathfrak {h}_{{\alpha ,2n}}^{\mathord {+}}\phi _{\mathord {\circ }}\left( {z}\right) +\mathfrak {b}_{{\alpha ,n+1}}\left( {z}\right) \psi _{\mathord {\circ }}\left( {z}\right) }\right)&\ne 0, \end{aligned}$$

(14.49)

$$\begin{aligned} \det \left( {\textbf{b}_{{{\mathord {\circ }},n}}\left( {z}\right) \mathfrak {h}_{{\alpha ,2n}}^{\mathord {+}}\phi _{\mathord {\circ }}\left( {z}\right) +\textbf{b}_{{{\mathord {\circ }},n+1}}\left( {z}\right) \psi _{\mathord {\circ }}\left( {z}\right) }\right)&\ne 0, \end{aligned}$$

(14.50)

and

$$\begin{aligned}{} & {} -\left[ {\mathfrak {a}_{{\alpha ,n}}\left( {z}\right) \mathfrak {h}_{{\alpha ,2n}}^{\mathord {+}}\phi _{\mathord {\circ }}\left( {z}\right) +\mathfrak {a}_{{\alpha ,n+1}}\left( {z}\right) \psi _{\mathord {\circ }}\left( {z}\right) } \right] \nonumber \\{} & {} \qquad \times \left[ {\mathfrak {b}_{{\alpha ,n}}\left( {z}\right) \mathfrak {h}_{{\alpha ,2n}}^{\mathord {+}}\phi _{\mathord {\circ }}\left( {z}\right) +\mathfrak {b}_{{\alpha ,n+1}}\left( {z}\right) \psi _{\mathord {\circ }}\left( {z}\right) } \right] ^{-1}\nonumber \\{} & {} \quad =-\left[ {\textbf{a}_{{{\mathord {\circ }},n}}\left( {z}\right) \mathfrak {h}_{{\alpha ,2n}}^{\mathord {+}}\phi _{\mathord {\circ }}\left( {z}\right) +\textbf{a}_{{{\mathord {\circ }},n+1}}\left( {z}\right) \psi _{\mathord {\circ }}\left( {z}\right) } \right] \nonumber \\{} & {} \qquad \times \left[ {\textbf{b}_{{{\mathord {\circ }},n}}\left( {z}\right) \mathfrak {h}_{{\alpha ,2n}}^{\mathord {+}}\phi _{\mathord {\circ }}\left( {z}\right) +\textbf{b}_{{{\mathord {\circ }},n+1}}\left( {z}\right) \psi _{\mathord {\circ }}\left( {z}\right) } \right] ^{-1}. \end{aligned}$$

(14.51)

Setting

$$\begin{aligned} S_{\mathord {\circ }}:=\phi _{\mathord {\circ }}\left( {w}\right) \quad \text {and}\quad T_{\mathord {\circ }}:=\psi _{\mathord {\circ }}\left( {w}\right) , \end{aligned}$$

(14.52)

we see, in view of (14.49), (14.50), and (14.51), that

$$\begin{aligned} {\mathfrak {R}}_{\mathord {\circ }}:=\mathfrak {b}_{{\alpha ,n}}\left( {w}\right) \mathfrak {h}_{{\alpha ,2n}}^{\mathord {+}}S_{\mathord {\circ }}+\mathfrak {b}_{{\alpha ,n+1}}\left( {w}\right) T_{\mathord {\circ }}\,\text {and}\, {\textbf{R}}_{\mathord {\circ }}:=\textbf{b}_{{{\mathord {\circ }},n}}\left( {w}\right) \mathfrak {h}_{{\alpha ,2n}}^{\mathord {+}}S_{\mathord {\circ }}+\textbf{b}_{{{\mathord {\circ }},n+1}}\left( {w}\right) T_{\mathord {\circ }}\nonumber \\ \end{aligned}$$

(14.53)

satisfy $\det {\mathfrak {R}}_{\mathord {\circ }}\ne 0$ and $\det {\textbf{R}}_{\mathord {\circ }}\ne 0$ as well as

$$\begin{aligned}{} & {} -\left[ {\mathfrak {a}_{{\alpha ,n}}\left( {w}\right) \mathfrak {h}_{{\alpha ,2n}}^{\mathord {+}}S_{\mathord {\circ }}+\mathfrak {a}_{{\alpha ,n+1}}\left( {w}\right) T_{\mathord {\circ }}} \right] {\mathfrak {R}}_{\mathord {\circ }}^{-1}\nonumber \\{} & {} \quad =-\left[ {\textbf{a}_{{{\mathord {\circ }},n}}\left( {w}\right) \mathfrak {h}_{{\alpha ,2n}}^{\mathord {+}}S_{\mathord {\circ }}+\textbf{a}_{{{\mathord {\circ }},n+1}}\left( {w}\right) T_{\mathord {\circ }}} \right] {\textbf{R}}_{\mathord {\circ }}^{-1}. \end{aligned}$$

(14.54)

Now we are going to justify that all assumptions for the application of Proposition 6.11 to the sequence $(a_j)_{j=0}^{2n+1}$ are satisfied. Because of $(a_j)_{j=0}^{\infty }\in \mathcal {H}^\succcurlyeq _{q,\infty }$, we have $(a_j)_{j=0}^{2n+1}\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,2n+1}$. According to (14.48), we have $P_{\mathord {\circ }}=\mathbb {P}_{\mathcal {R}\left( {\mathfrak {h}_{{\alpha ,2n}}}\right) }$ and $Q_{\mathord {\circ }}=\mathbb {P}_{\mathcal {N}\left( {\mathfrak {h}_{{\alpha ,2n}}}\right) }$. In view of $\Im w\in (0,\infty )$ and (14.52), from (14.45) and (14.46), we see $\Im \left( {w}\right) \Im \left( {T_{\mathord {\circ }}^*S_{\mathord {\circ }}}\right) \in \mathbb {C}_\succcurlyeq ^{{q\times q}}$ as well as

$$\begin{aligned} P_{\mathord {\circ }}S_{\mathord {\circ }}=S_{\mathord {\circ }},\quad S_{\mathord {\circ }}P_{\mathord {\circ }}=S_{\mathord {\circ }},\quad \quad \text {and}\quad T_{\mathord {\circ }}P_{\mathord {\circ }}=T_{\mathord {\circ }}-Q_{\mathord {\circ }}. \end{aligned}$$

(14.55)

Using additionally (14.53) and $\det {\mathfrak {R}}_{\mathord {\circ }}\ne 0$, we can thus apply Proposition 6.11 to infer that the matrix

$$\begin{aligned}{} & {} {\textbf{K}}_{\mathord {\circ }}:=\sqrt{\left( {\Im w}\right) ^{-1}\Im \chi _{{\alpha ,2n+1}}\left( {w}\right) }^{\mathord {+}}\left( {S_{\mathord {\circ }}+\left[ {\chi _{{\alpha ,2n+1}}\left( {w}\right) } \right] ^*T_{\mathord {\circ }}}\right) \nonumber \\{} & {} \quad \times \left( {S_{\mathord {\circ }}+\chi _{{\alpha ,2n+1}}\left( {w}\right) T_{\mathord {\circ }}}\right) ^{\mathord {+}}\sqrt{\left( {\Im w}\right) ^{-1}\Im \chi _{{\alpha ,2n+1}}\left( {w}\right) } \end{aligned}$$

(14.56)

fulfills

$$\begin{aligned}{} & {} -\left[ {\mathfrak {a}_{{\alpha ,n}}\left( {w}\right) \mathfrak {h}_{{\alpha ,2n}}^{\mathord {+}}S_{\mathord {\circ }}+\mathfrak {a}_{{\alpha ,n+1}}\left( {w}\right) T_{\mathord {\circ }}} \right] {\mathfrak {R}}_{\mathord {\circ }}^{-1}\nonumber \\{} & {} \quad =\mathscr {C}_{{\alpha ,2n+1}}\left( {w}\right) +\left( {w-\overline{w}}\right) ^{-1}\mathscr {A}_{{\alpha ,2n+1}}\left( {w}\right) {\textbf{K}}_{\mathord {\circ }}\mathscr {B}_{{\alpha ,2n+1}}\left( {w}\right) . \end{aligned}$$

(14.57)

By virtue of (14.56), (14.42), (14.52), and (14.47), we discern

$$\begin{aligned} {\textbf{K}}_{\mathord {\circ }}=\sqrt{B_{\mathord {\circ }}}^{\mathord {+}}\left[ {\phi _{\mathord {\circ }}\left( {w}\right) -E_{\mathord {\circ }}\psi _{\mathord {\circ }}\left( {w}\right) } \right] \left[ {\phi _{\mathord {\circ }}\left( {w}\right) -E_{\mathord {\circ }}^*\psi _{\mathord {\circ }}\left( {w}\right) } \right] ^{\mathord {+}}\sqrt{B_{\mathord {\circ }}} =P_{\mathord {\circ }}C_{\mathord {\circ }}P. \end{aligned}$$

(14.58)

The application of Lemma 6.9 to the sequence $(a_j)_{j=0}^{\infty }\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,\infty }$ yields

$$\begin{aligned} \mathscr {A}_{{\alpha ,2n}}\left( {w}\right) =\mathscr {A}_{{\alpha ,2n+1}}\left( {w}\right) ,\; \mathscr {B}_{{\alpha ,2n}}\left( {w}\right) =\mathscr {B}_{{\alpha ,2n+1}}\left( {w}\right) ,\; \mathscr {C}_{{\alpha ,2n}}\left( {w}\right) =\mathscr {C}_{{\alpha ,2n+1}}\left( {w}\right) . \nonumber \\ \end{aligned}$$

(14.59)

Regarding $(a_j)_{j=0}^{\infty }\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,\infty }$, from Proposition 6.10 we get $\mathcal {N}\left( {\mathscr {A}_{{\alpha ,2n}}\left( {w}\right) }\right) =\mathcal {N}\left( {\mathfrak {h}_{{\alpha ,2n}}}\right)$ and $\mathcal {R}\left( {\mathscr {B}_{{\alpha ,2n}}\left( {w}\right) }\right) =\mathcal {R}\left( {\mathfrak {h}_{{\alpha ,2n}}}\right)$. According to (14.48), we have $P_{\mathord {\circ }}=\mathbb {P}_{\mathcal {R}\left( {\mathfrak {h}_{{\alpha ,2n}}}\right) }$ and $Q_{\mathord {\circ }}=\mathbb {P}_{\mathcal {N}\left( {\mathfrak {h}_{{\alpha ,2n}}}\right) }$. Clearly, then $P_{\mathord {\circ }}\mathscr {B}_{{\alpha ,2n}}\left( {w}\right) =\mathscr {B}_{{\alpha ,2n}}\left( {w}\right)$ follows. By virtue of Remark A.8, we see furthermore $\mathcal {R}\left( {P_{\mathord {\circ }}}\right) =\mathcal {R}\left( {\mathfrak {h}_{{\alpha ,2n}}}\right)$ and $\mathcal {R}\left( {Q_{\mathord {\circ }}}\right) =\mathcal {N}\left( {\mathfrak {h}_{{\alpha ,2n}}}\right)$. Hence, we get $\mathscr {A}_{{\alpha ,2n}}\left( {w}\right) Q_{\mathord {\circ }}=O_{{q\times q}}$. Remark A.7 provides $\mathcal {N}\left( {\mathfrak {h}_{{\alpha ,2n}}^*}\right) =\mathcal {R}\left( {\mathfrak {h}_{{\alpha ,2n}}}\right) ^\bot$. Since, in view of $(a_j)_{j=0}^{\infty }\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,\infty }$, Remark 5.3 yields $\mathfrak {h}_{{\alpha ,2n}}^*=\mathfrak {h}_{{\alpha ,2n}}$, then $\mathcal {N}\left( {\mathfrak {h}_{{\alpha ,2n}}}\right) =\mathcal {R}\left( {\mathfrak {h}_{{\alpha ,2n}}}\right) ^\bot$ follows. Thus, Remark A.9 yields $P_{\mathord {\circ }}+Q_{\mathord {\circ }}=I_{q}$, implying $\mathscr {A}_{{\alpha ,2n}}\left( {w}\right) P_{\mathord {\circ }}=\mathscr {A}_{{\alpha ,2n}}\left( {w}\right)$. Consequently, we infer $\mathscr {A}_{{\alpha ,2n}}\left( {w}\right) P_{\mathord {\circ }}C_{\mathord {\circ }}P_{\mathord {\circ }}\mathscr {B}_{{\alpha ,2n}}\left( {w}\right) =\mathscr {A}_{{\alpha ,2n}}\left( {w}\right) C_{\mathord {\circ }}\mathscr {B}_{{\alpha ,2n}}\left( {w}\right)$. Using additionally (14.41), (14.59), (14.58), (14.57), and (14.54), we conclude

$$\begin{aligned}\begin{aligned} \eta X+s_{0}&=\mathscr {C}_{{\alpha ,2n}}\left( {w}\right) +\left( {w-\overline{w}}\right) ^{-1}\mathscr {A}_{{\alpha ,2n}}\left( {w}\right) C_{\mathord {\circ }}\mathscr {B}_{{\alpha ,2n}}\left( {w}\right) \\&=\mathscr {C}_{{\alpha ,2n}}\left( {w}\right) +\left( {w-\overline{w}}\right) ^{-1}\mathscr {A}_{{\alpha ,2n}}\left( {w}\right) P_{\mathord {\circ }}C_{\mathord {\circ }}P_{\mathord {\circ }}\mathscr {B}_{{\alpha ,2n}}\left( {w}\right) \\&=\mathscr {C}_{{\alpha ,2n+1}}\left( {w}\right) +\left( {w-\overline{w}}\right) ^{-1}\mathscr {A}_{{\alpha ,2n+1}}\left( {w}\right) {\textbf{K}}_{\mathord {\circ }}\mathscr {B}_{{\alpha ,2n+1}}\left( {w}\right) \\&=-\left[ {\mathfrak {a}_{{\alpha ,n}}\left( {w}\right) \mathfrak {h}_{{\alpha ,2n}}^{\mathord {+}}S_{\mathord {\circ }}+\mathfrak {a}_{{\alpha ,n+1}}\left( {w}\right) T_{\mathord {\circ }}} \right] {\mathfrak {R}}_{\mathord {\circ }}^{-1}\\&=-\left[ {\textbf{a}_{{{\mathord {\circ }},n}}\left( {w}\right) \mathfrak {h}_{{\alpha ,2n}}^{\mathord {+}}S_{\mathord {\circ }}+\textbf{a}_{{{\mathord {\circ }},n+1}}\left( {w}\right) T_{\mathord {\circ }}} \right] {\textbf{R}}_{\mathord {\circ }}^{-1}. \end{aligned}\end{aligned}$$

Using additionally $\det {\textbf{R}}_{\mathord {\circ }}\ne 0$, (14.53), (14.19), and Lemma 14.2(b), we then obtain

$$\begin{aligned} \begin{bmatrix} \eta X+s_{0}\\ I_{q}\end{bmatrix}= & {} \begin{bmatrix} -\left[ {\textbf{a}_{{{\mathord {\circ }},n}}\left( {w}\right) \mathfrak {h}_{{\alpha ,2n}}^{\mathord {+}}S_{\mathord {\circ }}+\textbf{a}_{{{\mathord {\circ }},n+1}}\left( {w}\right) T_{\mathord {\circ }}} \right] {\textbf{R}}_{\mathord {\circ }}^{-1}\\ {\textbf{R}}_{\mathord {\circ }}{\textbf{R}}_{\mathord {\circ }}^{-1}\end{bmatrix}\nonumber \\{} & {} \quad = \begin{bmatrix} -\textbf{a}_{{{\mathord {\circ }},n}}\left( {w}\right) \mathfrak {h}_{{\alpha ,2n}}^{\mathord {+}}S_{\mathord {\circ }}-\textbf{a}_{{{\mathord {\circ }},n+1}}\left( {w}\right) T_{\mathord {\circ }}\\ \textbf{b}_{{{\mathord {\circ }},n}}\left( {w}\right) \mathfrak {h}_{{\alpha ,2n}}^{\mathord {+}}S_{\mathord {\circ }}+\textbf{b}_{{{\mathord {\circ }},n+1}}\left( {w}\right) T_{\mathord {\circ }}\end{bmatrix} {\textbf{R}}_{\mathord {\circ }}^{-1}\nonumber \\{} & {} \quad = \begin{bmatrix} -\textbf{a}_{{{\mathord {\circ }},n}}\left( {w}\right) \mathfrak {k}_{2n+1}^{\mathord {+}}&{}-\textbf{a}_{{{\mathord {\circ }},n+1}}\left( {w}\right) \\ \textbf{b}_{{{\mathord {\circ }},n}}\left( {w}\right) \mathfrak {k}_{2n+1}^{\mathord {+}}&{}\textbf{b}_{{{\mathord {\circ }},n+1}}\left( {w}\right) \end{bmatrix} \begin{bmatrix} S_{\mathord {\circ }}\\ T_{\mathord {\circ }}\end{bmatrix} {\textbf{R}}_{\mathord {\circ }}^{-1}\nonumber \\{} & {} \quad =\mathfrak {V}^{\left( {\alpha }\right) }_{2n+2}\left( {w}\right) \begin{bmatrix} S_{\mathord {\circ }}\\ T_{\mathord {\circ }}\end{bmatrix} {\textbf{R}}_{\mathord {\circ }}^{-1}. \end{aligned}$$

(14.60)

Part 3: Obviously, the matrix ${\mathbb {D}}:=\bigl [{\begin{matrix}\eta I_{q}&{}\mathfrak {k}_{2n+3}\\ O_{{q\times q}}&{}I_{q}\end{matrix}}\bigr ]$ is invertible with

From (14.32) and (14.25), we see ${{\,\textrm{rank}\,}}\bigl [{\begin{matrix}S\\ T\end{matrix}}\bigr ]=q$. Consequently, the ${q\times q}$ block representation

$$\begin{aligned} {\mathbb {D}}^{-1}\begin{bmatrix} S\\ T\end{bmatrix}{\textbf{R}}^{-1}=\begin{bmatrix}Y\\ Z\end{bmatrix} \end{aligned}$$

(14.62)

of ${\mathbb {D}}^{-1}\bigl [{\begin{matrix}S\\ T\end{matrix}}\bigr ]{\textbf{R}}^{-1}$ fulfills ${{\,\textrm{rank}\,}}\bigl [{\begin{matrix}Y\\ Z\end{matrix}}\bigr ]={{\,\textrm{rank}\,}}\bigl [{\begin{matrix}S\\ T\end{matrix}}\bigr ]=q$. Lemma 14.5 shows $\mathfrak {V}^{\left( {\alpha }\right) }_{2n+3}\left( {w}\right) {\mathbb {D}}=\mathring{\mathbb {V}}_{2n+2}\left( {w}\right)$. Hence, using additionally (14.62) and (14.40), we deduce

$$\begin{aligned} \mathring{\mathbb {V}}_{2n+2}\left( {w}\right) \begin{bmatrix}Y\\ Z\end{bmatrix} =\mathfrak {V}^{\left( {\alpha }\right) }_{2n+3}\left( {w}\right) \begin{bmatrix} S\\ T\end{bmatrix}{\textbf{R}}^{-1}=\begin{bmatrix}X\\ I_{q}\end{bmatrix}. \end{aligned}$$

(14.63)

In view of (14.28), (14.19), and Remark A.10, we can infer

$$\begin{aligned} P=\mathbb {P}_{\mathcal {R}\left( {\mathfrak {k}_{2n+2}}\right) } =\mathfrak {k}_{2n+2}\mathfrak {k}_{2n+2}^{\mathord {+}}. \end{aligned}$$

(14.64)

Lemma 14.4 then shows

$$\begin{aligned} \eta \mathfrak {V}^{\left( {\alpha }\right) }_{2n+2}\left( {w}\right) \begin{bmatrix}P&{}O_{{q\times q}}\\ \mathfrak {k}_{2n+2}^{\mathord {+}}&{}I_{q}\end{bmatrix} =\begin{bmatrix}\eta I_{q}&{}s_{0}\\ O_{{q\times q}}&{}I_{q}\end{bmatrix}\mathring{\mathbb {V}}_{2n+2}\left( {w}\right) . \end{aligned}$$

Consequently, applying additionally (14.60) and (14.63), we conclude

$$\begin{aligned} \begin{aligned}&\mathfrak {V}^{\left( {\alpha }\right) }_{2n+2}\left( {w}\right) \begin{bmatrix}S_{\mathord {\circ }}\\ T_{\mathord {\circ }}\end{bmatrix}{\textbf{R}}_{\mathord {\circ }}^{-1}= \begin{bmatrix} \eta X+s_{0}\\ I_{q}\end{bmatrix} = \begin{bmatrix} \eta I_{q}&{}s_{0}\\ O_{{q\times q}}&{}I_{q}\end{bmatrix} \begin{bmatrix}X\\ I_{q}\end{bmatrix}\\&= \begin{bmatrix} \eta I_{q}&{}s_{0}\\ O_{{q\times q}}&{}I_{q}\end{bmatrix} \mathring{\mathbb {V}}_{2n+2}\left( {w}\right) \begin{bmatrix}Y\\ Z\end{bmatrix} =\eta \mathfrak {V}^{\left( {\alpha }\right) }_{2n+2}\left( {w}\right) \begin{bmatrix}P&{}O_{{q\times q}}\\ \mathfrak {k}_{2n+2}^{\mathord {+}}&{}I_{q}\end{bmatrix} \begin{bmatrix}Y\\ Z\end{bmatrix}. \end{aligned}\end{aligned}$$

(14.65)

Part 4: Regarding (14.62) and (14.61), we have

$$\begin{aligned} Y =\eta ^{-1}\left( { S-\mathfrak {k}_{2n+3} T}\right) {\textbf{R}}^{-1}\quad \text {and}\quad Z=T{\textbf{R}}^{-1}. \end{aligned}$$

(14.66)

In view of $(s_j)_{j=0}^{\infty }\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,\infty ,\alpha }$, Remark 9.3 yields $\mathfrak {k}_{2n+2}\mathfrak {k}_{2n+2}^{\mathord {+}}\mathfrak {k}_{2n+3}=\mathfrak {k}_{2n+3}$. By virtue of (14.64), therefore $P\mathfrak {k}_{2n+3}=\mathfrak {k}_{2n+3}$ follows. Taking additionally into account (14.35), from the first identity in (14.66) we then obtain

$$\begin{aligned} PY =\eta ^{-1}\left( {PS-P\mathfrak {k}_{2n+3}T}\right) {\textbf{R}}^{-1}=\eta ^{-1}\left( {S-\mathfrak {k}_{2n+3}T}\right) {\textbf{R}}^{-1}=Y. \end{aligned}$$

(14.67)

Let $\textbf{T}^{\left( {\alpha }\right) }_{2n+2}:\mathbb {C}\rightarrow \mathbb {C}^{{q\times q}}$ be defined by (14.7). Then, according to Lemma 14.7, we get $\mathfrak {W}^{\left( {\alpha }\right) }_{2n+2}\left( {w}\right) \mathfrak {V}^{\left( {\alpha }\right) }_{2n+2}\left( {w}\right) =\Delta$, where

$$\begin{aligned} \Delta :={{\,\textrm{diag}\,}}\left( {\mathfrak {k}_{2n+1}\mathfrak {k}_{2n+1}^{\mathord {+}},\eta ^{n+1}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) +\mathfrak {k}_{2n+1}^{\mathord {+}}\mathfrak {k}_{2n+1}+\textbf{T}^{\left( {\alpha }\right) }_{2n+2}\left( {w}\right) }\right) . \end{aligned}$$

Consequently, in view of (14.65), we infer

$$\begin{aligned} \Delta \begin{bmatrix} S_{\mathord {\circ }}\\ T_{\mathord {\circ }}\end{bmatrix}{\textbf{R}}_{\mathord {\circ }}^{-1}=\eta \Delta \begin{bmatrix}P&{}O_{{q\times q}}\\ \mathfrak {k}_{2n+2}^{\mathord {+}}&{}I_{q}\end{bmatrix}\begin{bmatrix}Y\\ Z\end{bmatrix} =\eta \Delta \begin{bmatrix}PY\\ \mathfrak {k}_{2n+2}^{\mathord {+}}Y+Z\end{bmatrix}. \end{aligned}$$

Therefore,

$$\begin{aligned} \mathfrak {k}_{2n+1}\mathfrak {k}_{2n+1}^{\mathord {+}}S_{\mathord {\circ }}{\textbf{R}}_{\mathord {\circ }}^{-1}=\eta \mathfrak {k}_{2n+1}\mathfrak {k}_{2n+1}^{\mathord {+}}PY \end{aligned}$$

(14.68)

and

$$\begin{aligned}{} & {} \left[ {\eta ^{n+1}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) +\mathfrak {k}_{2n+1}^{\mathord {+}}\mathfrak {k}_{2n+1}+\textbf{T}^{\left( {\alpha }\right) }_{2n+2}\left( {w}\right) } \right] T_{\mathord {\circ }}{\textbf{R}}_{\mathord {\circ }}^{-1}\nonumber \\{} & {} \quad =\eta \left[ {\eta ^{n+1}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) +\mathfrak {k}_{2n+1}^{\mathord {+}}\mathfrak {k}_{2n+1}+\textbf{T}^{\left( {\alpha }\right) }_{2n+2}\left( {w}\right) } \right] \left( {\mathfrak {k}_{2n+2}^{\mathord {+}}Y+Z}\right) .\qquad \end{aligned}$$

(14.69)

In view of $(s_j)_{j=0}^{\infty }\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,\infty ,\alpha }$, Remark 9.3 yields

$$\begin{aligned} \mathfrak {k}_{\ell }^{\mathord {+}}\mathfrak {k}_{\ell }\mathfrak {k}_{m}^{\mathord {+}}&=\mathfrak {k}_{m}^{\mathord {+}}{} & {} \text {and}&\mathfrak {k}_{m}\mathfrak {k}_{\ell }^{\mathord {+}}\mathfrak {k}_{\ell }&=\mathfrak {k}_{m}&\text {for all }\ell ,m\in \mathbb {N}_0\text { with }\ell \le m. \end{aligned}$$

(14.70)

If $n\ge 1$, then from (14.7) and (14.70) we can see that

$$\begin{aligned} \textbf{T}^{\left( {\alpha }\right) }_{2n+2}\left( {w}\right) \mathfrak {k}_{2n+2}^{\mathord {+}}=\sum _{k=0}^{n-1}\eta ^{n-k}\left( {\mathfrak {k}_{2k+1}^{\mathord {+}}\mathfrak {k}_{2k+1}\mathfrak {k}_{2n+2}^{\mathord {+}}-\mathfrak {k}_{2k+2}^{\mathord {+}}\mathfrak {k}_{2k+2}\mathfrak {k}_{2n+2}^{\mathord {+}}}\right) =O_{{q\times q}}\end{aligned}$$

and, for each $m\in \left\{ {2n+1,2n+2} \right\}$, moreover,

$$\begin{aligned} \mathfrak {k}_{m}\textbf{T}^{\left( {\alpha }\right) }_{2n+2}\left( {w}\right) =\sum _{k=0}^{n-1}\eta ^{n-k}\left( {\mathfrak {k}_{m}\mathfrak {k}_{2k+1}^{\mathord {+}}\mathfrak {k}_{2k+1}-\mathfrak {k}_{m}\mathfrak {k}_{2k+2}^{\mathord {+}}\mathfrak {k}_{2k+2}}\right) =O_{{q\times q}}\end{aligned}$$

are fulfilled. Regarding (14.7) also in the case $n=0$, we can conclude then in general

$$\begin{aligned} \textbf{T}^{\left( {\alpha }\right) }_{2n+2}\left( {w}\right) \mathfrak {k}_{2n+2}^{\mathord {+}}&=O_{{q\times q}},&\mathfrak {k}_{2n+1}\textbf{T}^{\left( {\alpha }\right) }_{2n+2}\left( {w}\right)&=O_{{q\times q}},&\mathfrak {k}_{2n+2}\textbf{T}^{\left( {\alpha }\right) }_{2n+2}\left( {w}\right)&=O_{{q\times q}}. \end{aligned}$$

(14.71)

Using (14.48), (14.19), and Remark A.10, we can infer

$$\begin{aligned} P_{\mathord {\circ }}=\mathfrak {k}_{2n+1}\mathfrak {k}_{2n+1}^{\mathord {+}}. \end{aligned}$$

(14.72)

Regarding additionally (14.20), the second identity in (14.71), (14.70), and (14.55), then

(14.73)

follows. In view of (14.64), we can infer from Remark 9.3 that

$$\begin{aligned} \mathfrak {k}_{\ell }\mathfrak {k}_{\ell }^{\mathord {+}}P=P\quad \text {for all }\ell \in \mathbb {Z}_{0,2n+2}. \end{aligned}$$

(14.74)

From (14.68) we thus can conclude

$$\begin{aligned} \mathfrak {k}_{2n+1}\mathfrak {k}_{2n+1}^{\mathord {+}}S_{\mathord {\circ }}{\textbf{R}}_{\mathord {\circ }}^{-1}=\eta \mathfrak {k}_{2n+1}\mathfrak {k}_{2n+1}^{\mathord {+}}PY =\eta PY. \end{aligned}$$

(14.75)

In view of (14.55) and (14.72), then

$$\begin{aligned} S_{\mathord {\circ }}{\textbf{R}}_{\mathord {\circ }}^{-1}= P_{\mathord {\circ }}S_{\mathord {\circ }}{\textbf{R}}_{\mathord {\circ }}^{-1}=\mathfrak {k}_{2n+1}\mathfrak {k}_{2n+1}^{\mathord {+}}S_{\mathord {\circ }}{\textbf{R}}_{\mathord {\circ }}^{-1}=\eta PY \end{aligned}$$

follows. Furthermore, (14.69), (14.70), and the first identity in (14.71) yield

$$\begin{aligned}{} & {} \left[ {\eta ^{n+1}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) +\mathfrak {k}_{2n+1}^{\mathord {+}}\mathfrak {k}_{2n+1}+\textbf{T}^{\left( {\alpha }\right) }_{2n+2}\left( {w}\right) } \right] T_{\mathord {\circ }}{\textbf{R}}_{\mathord {\circ }}^{-1}\nonumber \\{} & {} \quad =\eta \left[ {\eta ^{n+1}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) +\mathfrak {k}_{2n+1}^{\mathord {+}}\mathfrak {k}_{2n+1}+\textbf{T}^{\left( {\alpha }\right) }_{2n+2}\left( {w}\right) } \right] \left( {\mathfrak {k}_{2n+2}^{\mathord {+}}Y+Z}\right) \nonumber \\{} & {} \quad =\eta ^{n+2}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) \mathfrak {k}_{2n+2}^{\mathord {+}}Y+\eta \mathfrak {k}_{2n+1}^{\mathord {+}}\mathfrak {k}_{2n+1}\mathfrak {k}_{2n+2}^{\mathord {+}}Y+\eta \textbf{T}^{\left( {\alpha }\right) }_{2n+2}\left( {w}\right) \mathfrak {k}_{2n+2}^{\mathord {+}}Y\nonumber \\{} & {} \qquad +\eta \left[ {\eta ^{n+1}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) +\mathfrak {k}_{2n+1}^{\mathord {+}}\mathfrak {k}_{2n+1}+\textbf{T}^{\left( {\alpha }\right) }_{2n+2}\left( {w}\right) } \right] Z\nonumber \\{} & {} \quad =\eta ^{n+2}\left( {\mathfrak {k}_{2n+2}^{\mathord {+}}-\mathfrak {k}_{2n+2}^{\mathord {+}}}\right) Y+\eta \mathfrak {k}_{2n+2}^{\mathord {+}}Y\nonumber \\{} & {} \qquad +\eta \left[ {\eta ^{n+1}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) +\mathfrak {k}_{2n+1}^{\mathord {+}}\mathfrak {k}_{2n+1}+\textbf{T}^{\left( {\alpha }\right) }_{2n+2}\left( {w}\right) } \right] Z\nonumber \\{} & {} \quad =\eta \left( {\mathfrak {k}_{2n+2}^{\mathord {+}}Y+\left[ {\eta ^{n+1}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) +\mathfrak {k}_{2n+1}^{\mathord {+}}\mathfrak {k}_{2n+1}+\textbf{T}^{\left( {\alpha }\right) }_{2n+2}\left( {w}\right) } \right] Z}\right) . \end{aligned}$$

(14.76)

In view of (14.20), we infer from Remark A.12 that $\left( {\mathfrak {k}_{2n+2}^{\mathord {+}}}\right) ^*=\mathfrak {k}_{2n+2}^{\mathord {+}}$ and $\left( {\mathfrak {k}_{j}^{\mathord {+}}\mathfrak {k}_{j}}\right) ^*=\mathfrak {k}_{j}\mathfrak {k}_{j}^{\mathord {+}}$ for all $j\in \mathbb {N}_0$. Regarding (14.74), in particular $\left( {\mathfrak {k}_{2n+1}^{\mathord {+}}\mathfrak {k}_{2n+1}}\right) ^*P=P$ and $\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) ^*P=O_{{q\times q}}$ follow. Using (14.64), (14.20), and the last identity in (14.71), we obtain $\left[ {\textbf{T}^{\left( {\alpha }\right) }_{2n+2}\left( {w}\right) } \right] ^*P=\left[ {\mathfrak {k}_{2n+2}\textbf{T}^{\left( {\alpha }\right) }_{2n+2}\left( {w}\right) } \right] ^*\mathfrak {k}_{2n+2}^{\mathord {+}}=O_{{q\times q}}$. Taking additionally into account (14.76), (14.75), and (14.67), we then get

$$\begin{aligned} \begin{aligned}&\left( {\left[ {\eta ^{n+1}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) +\mathfrak {k}_{2n+1}^{\mathord {+}}\mathfrak {k}_{2n+1} +\textbf{T}^{\left( {\alpha }\right) }_{2n+2}\left( {w}\right) } \right] T_{\mathord {\circ }}{\textbf{R}}_{\mathord {\circ }}^{-1}}\right) ^*\left( {\mathfrak {k}_{2n+1}\mathfrak {k}_{2n+1}^{\mathord {+}}S_{\mathord {\circ }}{\textbf{R}}_{\mathord {\circ }}^{-1}}\right) \\&=\overline{\eta }\left( {\mathfrak {k}_{2n+2}^{\mathord {+}}Y+\left[ {\eta ^{n+1}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) +\mathfrak {k}_{2n+1}^{\mathord {+}}\mathfrak {k}_{2n+1}+\textbf{T}^{\left( {\alpha }\right) }_{2n+2}\left( {w}\right) } \right] Z}\right) ^*\left( {\eta PY}\right) \\&=|{\eta } |^2Y^*\left( {\mathfrak {k}_{2n+2}^{\mathord {+}}}\right) ^*PY\\&\qquad +|{\eta } |^2Z^*\left[ {\eta ^{n+1}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) +\mathfrak {k}_{2n+1}^{\mathord {+}}\mathfrak {k}_{2n+1}+\textbf{T}^{\left( {\alpha }\right) }_{2n+2}\left( {w}\right) } \right] ^*PY\\&=|{\eta } |^2Y^*\left( {\mathfrak {k}_{2n+2}^{\mathord {+}}}\right) ^*PY+|{\eta } |^2\overline{\eta } ^{n+1}Z^*\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) ^*PY\\&\qquad +|{\eta } |^2Z^*\left( {\mathfrak {k}_{2n+1}^{\mathord {+}}\mathfrak {k}_{2n+1}}\right) ^*PY+|{\eta } |^2Z^*\left[ {\textbf{T}^{\left( {\alpha }\right) }_{2n+2}\left( {w}\right) } \right] ^*PY\\&=|{\eta } |^2Y^*\mathfrak {k}_{2n+2}^{\mathord {+}}PY+|{\eta } |^2Z^*PY =|{\eta } |^2\left( {Y^*\mathfrak {k}_{2n+2}^{\mathord {+}}Y+Z^*Y}\right) . \end{aligned}\end{aligned}$$

(14.77)

Combining (14.73) and (14.77), we obtain ${\textbf{R}}_{\mathord {\circ }}^{-*}T_{\mathord {\circ }}^*S_{\mathord {\circ }}{\textbf{R}}_{\mathord {\circ }}^{-1}=|{\eta } |^2\left( {Y^*\mathfrak {k}_{2n+2}^{\mathord {+}}Y+Z^*Y}\right)$ and, thus, we conclude $Z^*Y=|{\eta } |^{-2}{\textbf{R}}_{\mathord {\circ }}^{-*}T_{\mathord {\circ }}^*S_{\mathord {\circ }}{\textbf{R}}_{\mathord {\circ }}^{-1}-Y^*\mathfrak {k}_{2n+2}^{\mathord {+}}Y$. Since $\left( {\mathfrak {k}_{2n+2}^{\mathord {+}}}\right) ^*=\mathfrak {k}_{2n+2}^{\mathord {+}}$ implies $\Im \left( {Y^*\mathfrak {k}_{2n+2}^{\mathord {+}}Y}\right) =O_{{q\times q}}$ and (14.52) and (14.45) show $\Im \left( { T_{\mathord {\circ }}^*S_{\mathord {\circ }}}\right) \in \mathbb {C}_\succcurlyeq ^{{q\times q}}$, the application of Remarks A.1, A.2, and A.6 then yields

$$\begin{aligned} \Im \left( {Z^*Y}\right)= & {} |{\eta } |^{-2}\Im \left( { {\textbf{R}}_{\mathord {\circ }}^{-*}T_{\mathord {\circ }}^*S_{\mathord {\circ }}{\textbf{R}}_{\mathord {\circ }}^{-1}}\right) -\Im \left( {Y^*\mathfrak {k}_{2n+2}^{\mathord {+}}Y}\right) \nonumber \\= & {} |{\eta } |^{-2}{\textbf{R}}_{\mathord {\circ }}^{-*}\Im \left( { T_{\mathord {\circ }}^*S_{\mathord {\circ }}}\right) {\textbf{R}}_{\mathord {\circ }}^{-1}\in \mathbb {C}_\succcurlyeq ^{{q\times q}}. \end{aligned}$$

(14.78)

Because of (14.20), we have $\Im \left( {T^*\mathfrak {k}_{2n+3}T}\right) =O_{{q\times q}}$. From (14.32) and (14.25) we see $\Im \left( {T^*S}\right) \in \mathbb {C}_\succcurlyeq ^{{q\times q}}$. Hence, using (14.66) as well as Remarks A.2, A.1, and A.6, we infer

$$\begin{aligned}&\Im \left( {\eta Z^*Y}\right) =\Im \left( {\eta \left( { T{\textbf{R}}^{-1}}\right) ^*\left[ {\eta ^{-1}\left( { S-\mathfrak {k}_{2n+3} T}\right) {\textbf{R}}^{-1}} \right] }\right) \nonumber \\&=\Im \left( {{\textbf{R}}^{-*}T^*\left( { S-\mathfrak {k}_{2n+3} T}\right) {\textbf{R}}^{-1}}\right) \nonumber \\&={\textbf{R}}^{-*}\left[ {\Im \left( { T^*S}\right) -\Im \left( { T^*\mathfrak {k}_{2n+3} T}\right) } \right] {\textbf{R}}^{-1}={\textbf{R}}^{-*}\Im \left( { T^*S}\right) {\textbf{R}}^{-1}\in \mathbb {C}_\succcurlyeq ^{{q\times q}}. \end{aligned}$$

(14.79)

Part 5: Let $\pi ,\rho :\mathbb {C}\rightarrow \mathbb {C}^{{q\times q}}$ be defined by

$$\begin{aligned} \pi \left( {z}\right) :=Y \,\text {and}\, \rho \left( {z}\right) :=\frac{w-z}{\Im w}HY+Z, \,\text {where}\, H:=\left( {Y^{\mathord {+}}}\right) ^*\Im \left( {Z^*Y}\right) Y^{\mathord {+}}.\nonumber \\ \end{aligned}$$

(14.80)

In view of (14.78) and (14.80), we can apply Lemma 8.5 and it follows that $\pi$ and $\rho$ are both holomorphic in $\mathbb {C}$ fulfilling

$$\begin{aligned} \begin{bmatrix}\pi \left( {w}\right) \\ \rho \left( {w}\right) \end{bmatrix} =\begin{bmatrix}Y\\ Z\end{bmatrix} \end{aligned}$$

(14.81)

as well as

$$\begin{aligned} {{\,\textrm{rank}\,}}\begin{bmatrix}\pi \left( {z}\right) \\ \rho \left( {z}\right) \end{bmatrix}&={{\,\textrm{rank}\,}}\begin{bmatrix}Y\\ Z\end{bmatrix}, \end{aligned}$$

(14.82)

$$\begin{aligned} \Im \left( {\left[ {\rho \left( {z}\right) } \right] ^*\pi \left( {z}\right) }\right)&=\frac{\Im z}{\Im w}\Im \left( {Z^*Y}\right) , \end{aligned}$$

(14.83)

and

$$\begin{aligned} \Im \left( {\left( {z-\alpha }\right) \left[ {\rho \left( {z}\right) } \right] ^*\pi \left( {z}\right) }\right)&=\frac{\Im z}{\Im w}\Im \left( {\eta Z^*Y}\right) \end{aligned}$$

(14.84)

for all $z\in \mathbb {C}$. Let $\phi _{\mathord {\bullet }},\psi _{\mathord {\bullet }}:\mathbb {C}\backslash {[\alpha ,\infty )}\rightarrow \mathbb {C}^{{q\times q}}$ be defined by

$$\begin{aligned} \phi _{\mathord {\bullet }}\left( {z}\right)&:=\pi \left( {z}\right) \;\text {and}\; \psi _{\mathord {\bullet }}\left( {z}\right) :=\rho \left( {z}\right) . \end{aligned}$$

(14.85)

Clearly, $\mathcal {D}_{\mathord {\bullet }}:=\emptyset$ is a discrete subset of $\mathbb {C}\backslash {[\alpha ,\infty )}$. Keeping in mind that $\pi$ and $\rho$ are holomorphic in $\mathbb {C}$, we see that $\phi _{\mathord {\bullet }}$ and $\psi _{\mathord {\bullet }}$ are holomorphic and, in particular, meromorphic in $\mathbb {C}\backslash {[\alpha ,\infty )}$. For all $z\in \mathbb {C}\backslash {[\alpha ,\infty )}$, equations (14.85), (14.82), and ${{\,\textrm{rank}\,}}\bigl [{\begin{matrix}Y\\ Z\end{matrix}}\bigr ]=q$ imply ${{\,\textrm{rank}\,}}\bigl [{\begin{matrix}\phi _{\mathord {\bullet }}\left( {z}\right) \\ \psi _{\mathord {\bullet }}\left( {z}\right) \end{matrix}}\bigr ]={{\,\textrm{rank}\,}}\bigl [{\begin{matrix}\pi \left( {z}\right) \\ \rho \left( {z}\right) \end{matrix}}\bigr ]={{\,\textrm{rank}\,}}\left[ \begin{array}{c}Y\\ Z\end{array}\right] =q$. Regarding $\Im w\in (0,\infty )$, for all $z\in \mathbb {C}\backslash \mathbb {R}$, moreover (14.85), (14.83), and (14.78) yield

$$\begin{aligned} \left( {\Im z}\right) ^{-1}\Im \left( {\left[ {\psi _{\mathord {\bullet }}\left( {z}\right) } \right] ^*\phi _{\mathord {\bullet }}\left( {z}\right) }\right) =\left( {\Im z}\right) ^{-1}\Im \left( {\left[ {\rho \left( {z}\right) } \right] ^*\pi \left( {z}\right) }\right) =\left( {\Im w}\right) ^{-1}\Im \left( {Z^*Y}\right) \in \mathbb {C}_\succcurlyeq ^{{q\times q}}, \end{aligned}$$

whereas (14.85), (14.84), and (14.79) provide

$$\begin{aligned}\begin{aligned} \left( {\Im z}\right) ^{-1}\Im \left( {\left( {z-\alpha }\right) \left[ {\psi _{\mathord {\bullet }}\left( {z}\right) } \right] ^*\phi _{\mathord {\bullet }}\left( {z}\right) }\right)&=\left( {\Im z}\right) ^{-1}\Im \left( {\left( {z-\alpha }\right) \left[ {\rho \left( {z}\right) } \right] ^*\pi \left( {z}\right) }\right) \\&=\left( {\Im w}\right) ^{-1}\Im \left( {\eta Z^*Y}\right) \in \mathbb {C}_\succcurlyeq ^{{q\times q}}. \end{aligned}\end{aligned}$$

In view of Remark 8.2, then, according to Definition 8.1, the pair $\left( {\phi _{\mathord {\bullet }}};{\psi _{\mathord {\bullet }}}\right)$ belongs to $\mathcal{P}\mathcal{S}_{q}\left( {\mathbb {C}\backslash {[\alpha ,\infty )}}\right)$ and the set $\mathcal {D}_{\mathord {\bullet }}$ belongs to $\mathscr {D}_{\mathord {\bullet }}\left( {\phi _{\mathord {\bullet }},\psi _{\mathord {\bullet }}}\right)$. Because of Definition 5.2 and (5.2) we have $\mathfrak {h}_{2n+2}=L_{n+1}$. Using additionally (14.85), (14.28), (14.80), and (14.67), we can conclude $\mathbb {P}_{\mathcal {R}\left( {L_{n+1}}\right) }\phi _{\mathord {\bullet }}\left( {z}\right) =\mathbb {P}_{\mathcal {R}\left( {\mathfrak {h}_{2n+2}}\right) }\pi \left( {z}\right) =PY=Y=\pi \left( {z}\right) =\phi _{\mathord {\bullet }}\left( {z}\right)$ for all $z\in \mathbb {C}\backslash {[\alpha ,\infty )}$. By virtue of Notation 8.4, consequently $\left( {\phi _{\mathord {\bullet }}};{\psi _{\mathord {\bullet }}}\right) \in \mathcal {P}_{{\mathord {\bullet }},\alpha }\left[ {L_{n+1}} \right]$.

Part 6: From $\left( {\phi _{\mathord {\bullet }}};{\psi _{\mathord {\bullet }}}\right) \in \mathcal {P}_{{\mathord {\bullet }},\alpha }\left[ {L_{n+1}} \right]$, Theorem 10.17(a) and the notations therein, we see that $\det \left( {\tilde{\textbf{p}}_{2n+2}^\flat L_{n+1}^{\mathord {+}}\phi _{\mathord {\bullet }}+\tilde{\textbf{p}}_{2n+3}\psi _{\mathord {\bullet }}}\right)$ does not vanish identically and that

$$\begin{aligned} F :=-\left( {\tilde{\textbf{q}}_{2n+2}^\flat L_{n+1}^{\mathord {+}}\phi _{\mathord {\bullet }}+\tilde{\textbf{q}}_{2n+3}\psi _{\mathord {\bullet }}}\right) \left( {\tilde{\textbf{p}}_{2n+2}^\flat L_{n+1}^{\mathord {+}}\phi _{\mathord {\bullet }}+\tilde{\textbf{p}}_{2n+3}\psi _{\mathord {\bullet }}}\right) ^{-1}\end{aligned}$$

(14.86)

belongs to ${\mathcal {S}_{0,q}\left[ {\mathbb {C}\backslash {[\alpha ,\infty )};(s_j)_{j=0}^{2n+2},\preccurlyeq } \right] }$. Furthermore, regarding (14.63), (14.81), (14.85), and (14.3), we have

$$\begin{aligned}\begin{aligned} \begin{bmatrix}X\\ I_{q}\end{bmatrix}&=\mathring{\mathbb {V}}_{2n+2}\left( {w}\right) \begin{bmatrix}Y\\ Z\end{bmatrix} =\mathring{\mathbb {V}}_{2n+2}\left( {w}\right) \begin{bmatrix}\pi \left( {w}\right) \\ \rho \left( {w}\right) \end{bmatrix}\\&=\mathring{\mathbb {V}}_{2n+2}\left( {w}\right) \begin{bmatrix}\phi _{\mathord {\bullet }}\left( {w}\right) \\ \psi _{\mathord {\bullet }}\left( {w}\right) \end{bmatrix} = \begin{bmatrix} -\eta \textbf{q}_{2n+2}\left( {w}\right) \mathfrak {k}_{2n+2}^{\mathord {+}}&{}-\textbf{q}_{2n+3}\left( {w}\right) \\ \eta \textbf{p}_{2n+2}\left( {w}\right) \mathfrak {k}_{2n+2}^{\mathord {+}}&{}\textbf{p}_{2n+3}\left( {w}\right) \end{bmatrix}\begin{bmatrix}\phi _{\mathord {\bullet }}\left( {w}\right) \\ \psi _{\mathord {\bullet }}\left( {w}\right) \end{bmatrix}. \end{aligned}\end{aligned}$$

Since $\mathfrak {k}_{2n+2}=\mathfrak {h}_{2n+2}=L_{n+1}$, we consequently get

$$\begin{aligned}\begin{aligned} X&=-\eta \textbf{q}_{2n+2}\left( {w}\right) \mathfrak {k}_{2n+2}^{\mathord {+}}\phi _{\mathord {\bullet }}\left( {w}\right) -\textbf{q}_{2n+3}\left( {w}\right) \psi _{\mathord {\bullet }}\left( {w}\right) \\&=-\left[ {\tilde{\textbf{q}}_{2n+2}^\flat \left( {w}\right) L_{n+1}^{\mathord {+}}\phi _{\mathord {\bullet }}\left( {w}\right) +\tilde{\textbf{q}}_{2n+3}\left( {w}\right) \psi _{\mathord {\bullet }}\left( {w}\right) } \right] \end{aligned}\end{aligned}$$

and

$$\begin{aligned}\begin{aligned} I_{q}&=\eta \textbf{p}_{2n+2}\left( {w}\right) \mathfrak {k}_{2n+2}^{\mathord {+}}\phi _{\mathord {\bullet }}\left( {w}\right) +\textbf{p}_{2n+3}\left( {w}\right) \psi _{\mathord {\bullet }}\left( {w}\right) \\&=\tilde{\textbf{p}}_{2n+2}^\flat \left( {w}\right) L_{n+1}^{\mathord {+}}\phi _{\mathord {\bullet }}\left( {w}\right) +\tilde{\textbf{p}}_{2n+3}\left( {w}\right) \psi _{\mathord {\bullet }}\left( {w}\right) . \end{aligned}\end{aligned}$$

In particular, $\det \left( {\tilde{\textbf{p}}_{2n+2}^\flat \left( {w}\right) L_{n+1}^{\mathord {+}}\phi _{\mathord {\bullet }}\left( {w}\right) +\tilde{\textbf{p}}_{2n+3}\left( {w}\right) \psi _{\mathord {\bullet }}\left( {w}\right) }\right) \ne 0$. Taking additionally into account (14.86), we finally see $X=-\big [\tilde{\textbf{q}}_{2n+2}^\flat \left( {w}\right) L_{n+1}^{\mathord {+}}\phi _{\mathord {\bullet }}\left( {w}\right) +\tilde{\textbf{q}}_{2n+3}\left( {w}\right) \psi _{\mathord {\bullet }}\left( {w}\right) \big ]\left[ {\tilde{\textbf{p}}_{2n+2}^\flat \left( {w}\right) L_{n+1}^{\mathord {+}}\phi _{\mathord {\bullet }}\left( {w}\right) +\tilde{\textbf{p}}_{2n+3}\left( {w}\right) \psi _{\mathord {\bullet }}\left( {w}\right) } \right] ^{-1}=F\left( {w}\right)$. $\square$

Now we can summarize our results to obtain our first main result, which describes the set of possible values of the functions corresponding to the solutions of the Stieltjes moment problem in the case of an odd number of prescribed matrix moments.

Theorem 14.9

Let $n\in \mathbb {N}_0$, let $(s_j)_{j=0}^{2n+2}\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,2n+2,\alpha }$, and let $w\in \Pi _{\mathord {+}}$. Then

$$\begin{aligned} \left\{ {F\left( {w}\right) }:{F\in {\mathcal {S}_{0,q}\left[ {\mathbb {C}\backslash {[\alpha ,\infty )};(s_j)_{j=0}^{2n+2},\preccurlyeq } \right] }}\right\} =\mathscr {K}_{2n+2}\left( {w}\right) \cap \mathscr {K}_{{{\mathord {\circ }},2n}}\left( {w}\right) . \end{aligned}$$

Proof

Combine Propositions 13.4 and 14.8. $\square$

15 The Case of an Even Number of Prescribed Matrix Moments

After we discussed the case of an odd number of given matrix moments in Sect. 14, we now give a description of the possible values of the Stieltjes transforms of the solutions of the considered truncated Stieltjes moment problem in the case that an even number of matrix moments is prescribed.

Lemma 15.1

Suppose $\kappa \ge 1$. Let $(s_j)_{j=0}^{\kappa }\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,\kappa ,\alpha }$, let $n\in \mathbb {N}_0$ be such that $2n+1\le \kappa$ and let $z\in \mathbb {C}$. Then

$$\begin{aligned} \mathfrak {V}^{\left( {\alpha }\right) }_{2n+1}\left( {z}\right) \begin{bmatrix} \mathfrak {k}_{2n+1}\mathfrak {k}_{2n+1}^{\mathord {+}}&{}O_{{q\times q}}\\ \mathfrak {k}_{2n+1}^{\mathord {+}}&{}I_{q}\end{bmatrix} =\mathring{\mathbb {V}}_{2n+1}\left( {z}\right) . \end{aligned}$$

Proof

In view of Lemma 14.2(a) as well as Lemma 11.3 and (11.1), we have

Using (10.4) and Notation 14.3, then we get finally

$$\begin{aligned}\begin{aligned} \mathfrak {V}^{\left( {\alpha }\right) }_{2n+1} \begin{bmatrix} \mathfrak {k}_{2n+1}\mathfrak {k}_{2n+1}^{\mathord {+}}&{}O_{{q\times q}}\\ \mathfrak {k}_{2n+1}^{\mathord {+}}&{}I_{q}\end{bmatrix} = \begin{bmatrix} -\varepsilon _{2n+1}\textbf{q}_{2n+1}\mathfrak {k}_{2n+1}^{\mathord {+}}&{}-\textbf{q}_{2n+2}\\ \varepsilon _{2n+1}\textbf{p}_{2n+1}\mathfrak {k}_{2n+1}^{\mathord {+}}&{}\textbf{p}_{2n+2} \end{bmatrix} =\mathring{\mathbb {V}}_{2n+1}. \end{aligned}\end{aligned}$$

$\square$

Lemma 15.2

Suppose $\kappa \ge 2$. Let $(s_j)_{j=0}^{\kappa }\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,\kappa ,\alpha }$, let $n\in \mathbb {N}_0$ be such that $2n+2\le \kappa$, and let $z\in \mathbb {C}$. Then

$$\begin{aligned} \mathfrak {V}^{\left( {\alpha }\right) }_{2n+2}\left( {z}\right) \begin{bmatrix} \left( {z-\alpha }\right) I_{q}&{}\mathfrak {k}_{2n+2}\\ O_{{q\times q}}&{}I_{q}\end{bmatrix} = \begin{bmatrix} \left( {z-\alpha }\right) I_{q}&{}s_{0}\\ O_{{q\times q}}&{} I_{q}\end{bmatrix} \mathring{\mathbb {V}}_{2n+1}\left( {z}\right) . \end{aligned}$$

Proof

Let $\zeta :=z-\alpha$. Using Lemma 14.2(b) as well as Remarks 12.7 and 12.6, we obtain

In view of (14.4), Lemmata 12.3 and 11.3, (12.1), and (11.1), we get furthermore

Hence,

Consequently, the asserted identity follows. $\square$

Lemma 15.3

Suppose $\kappa \ge 1$. Let $(s_j)_{j=0}^{\kappa }\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,\kappa ,\alpha }$. For each $n\in \mathbb {N}_0$ fulfilling $2n+1\le \kappa$, let $\textbf{T}^{\left( {\alpha }\right) }_{2n+1}:\mathbb {C}\rightarrow \mathbb {C}^{{q\times q}}$ be defined by

$$\begin{aligned} \textbf{T}^{\left( {\alpha }\right) }_{2n+1}\left( {z}\right) :={\left\{ \begin{array}{ll} O_{{q\times q}},&{}\text { if }n=0\\ {\sum }_{\ell =0}^{n-1}\left( {z-\alpha }\right) ^{n-\ell }\left( {\mathfrak {k}_{2\ell }^{\mathord {+}}\mathfrak {k}_{2\ell }-\mathfrak {k}_{2\ell +1}^{\mathord {+}}\mathfrak {k}_{2\ell +1}}\right) ,&{}\text { if }n\ge 1 \end{array}\right. }. \end{aligned}$$

(15.1)

For each $n\in \mathbb {N}_0$ fulfilling $2n+1\le \kappa$ and all $z\in \mathbb {C}$, then

$$\begin{aligned}{} & {} \mathfrak {W}^{\left( {\alpha }\right) }_{2n+1}\left( {z}\right) \mathfrak {V}^{\left( {\alpha }\right) }_{2n+1}\left( {z}\right) \nonumber \\{} & {} \quad ={{\,\textrm{diag}\,}}\left( {\mathfrak {k}_{2n}\mathfrak {k}_{2n}^{\mathord {+}},\left( {z-\alpha }\right) ^{n+1}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) +\mathfrak {k}_{2n}^{\mathord {+}}\mathfrak {k}_{2n}+\textbf{T}^{\left( {\alpha }\right) }_{2n+1}\left( {z}\right) }\right) . \end{aligned}$$

(15.2)

Proof

Let $z\in \mathbb {C}$ and let $\zeta :=z-\alpha$. First note that Remark 9.3 yields (9.3) and (9.4). Our proof uses mathematical induction and is divided into three parts.

Part 1: Because of Notations 14.6 and 14.1, (9.3), and (9.4), we have

(15.3)

Regarding (15.1), hence (15.2) is checked for $n=0$. In particular, the proof is complete in the case $\kappa \le 2$.

Part 2: Assume $\kappa \ge 3$. In view of Notation 14.1, (15.3), (9.3), and (9.4), we see that

(15.4)

holds true. Due to (9.4), yielding

$$\begin{aligned}\begin{aligned}&\left[ {\zeta \left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) +\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}} \right] \left[ {\zeta I_{q}-\left( {\mathfrak {k}_{1}^{\mathord {+}}\mathfrak {k}_{2}+\mathfrak {k}_{2}^{\mathord {+}}\mathfrak {k}_{3}}\right) } \right] \\&=\zeta ^2\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) -\zeta \left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) \left( {\mathfrak {k}_{1}^{\mathord {+}}\mathfrak {k}_{2}+\mathfrak {k}_{2}^{\mathord {+}}\mathfrak {k}_{3}}\right) +\zeta \mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}\left( {\mathfrak {k}_{1}^{\mathord {+}}\mathfrak {k}_{2}+\mathfrak {k}_{2}^{\mathord {+}}\mathfrak {k}_{3}}\right) \\&=\zeta ^2\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) +\zeta \mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}-\left( {\mathfrak {k}_{1}^{\mathord {+}}\mathfrak {k}_{2}+\mathfrak {k}_{2}^{\mathord {+}}\mathfrak {k}_{3}}\right) , \end{aligned}\end{aligned}$$

equation (15.4) can be further simplified to

Using additionally Notation 14.6, (9.3), (9.4), and (15.1) we consequently obtain

(15.5)

Hence, (15.2) is checked for $n=1$. In particular, the proof is complete in the case $\kappa \le 4$.

Part 3: Now assume $\kappa \ge 5$. Then, in view of (15.5), there exists an integer $k \in \mathbb {N}$ with $2\left( { k +1}\right) +1\le \kappa$ such that (15.2) is fulfilled for $n= k$. Taking additionally into account Notation 14.1 and (9.3), then

(15.6)

follows, where

$$\begin{aligned} L :=\left[ {\zeta ^{ k +1}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) +\mathfrak {k}_{2 k }^{\mathord {+}}\mathfrak {k}_{2 k }+\textbf{T}^{\left( {\alpha }\right) }_{2k+1}\left( {z}\right) } \right] \mathfrak {k}_{2 k +2}^{\mathord {+}}\end{aligned}$$

(15.7)

and

$$\begin{aligned}{} & {} R :=\left[ {\zeta ^{ k +1}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) +\mathfrak {k}_{2 k }^{\mathord {+}}\mathfrak {k}_{2 k }+\textbf{T}^{\left( {\alpha }\right) }_{2k+1}\left( {z}\right) } \right] \nonumber \\{} & {} \quad \times \left( {\zeta I_{q}-\mathfrak {k}_{2k+1}^{\mathord {+}}\mathfrak {k}_{2 k +2}-\mathfrak {k}_{2 k +2}^{\mathord {+}}\mathfrak {k}_{2 k +3}}\right) . \end{aligned}$$

(15.8)

Due to (15.7), (9.4), and (15.1), we have

$$\begin{aligned} \begin{aligned} L&=\zeta ^{ k +1}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) \mathfrak {k}_{2 k +2}^{\mathord {+}}+\mathfrak {k}_{2 k }^{\mathord {+}}\mathfrak {k}_{2 k }\mathfrak {k}_{2 k +2}^{\mathord {+}}+\textbf{T}^{\left( {\alpha }\right) }_{2k+1}\left( {z}\right) \mathfrak {k}_{2 k +2}^{\mathord {+}}\\&=\mathfrak {k}_{2 k +2}^{\mathord {+}}+\sum _{\ell =0}^{k-1}\zeta ^{k-\ell }\left( {\mathfrak {k}_{2\ell }^{\mathord {+}}\mathfrak {k}_{2\ell }-\mathfrak {k}_{2\ell +1}^{\mathord {+}}\mathfrak {k}_{2\ell +1}}\right) \mathfrak {k}_{2 k +2}^{\mathord {+}}=\mathfrak {k}_{2 k +2}^{\mathord {+}}, \end{aligned}\end{aligned}$$

(15.9)

whereas from (15.8), (9.4), and (15.1), we conclude

$$\begin{aligned} \begin{aligned}&R =\zeta ^{ k +2}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) +\zeta \mathfrak {k}_{2 k }^{\mathord {+}}\mathfrak {k}_{2 k }+\zeta \textbf{T}^{\left( {\alpha }\right) }_{2k+1}\left( {z}\right) \\&\qquad -\zeta ^{ k +1}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) \mathfrak {k}_{2k+1}^{\mathord {+}}\mathfrak {k}_{2 k +2}-\mathfrak {k}_{2 k }^{\mathord {+}}\mathfrak {k}_{2 k }\mathfrak {k}_{2k+1}^{\mathord {+}}\mathfrak {k}_{2 k +2}-\textbf{T}^{\left( {\alpha }\right) }_{2k+1}\left( {z}\right) \mathfrak {k}_{2k+1}^{\mathord {+}}\mathfrak {k}_{2 k +2}\\&\qquad -\zeta ^{ k +1}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) \mathfrak {k}_{2 k +2}^{\mathord {+}}\mathfrak {k}_{2 k +3}-\mathfrak {k}_{2 k }^{\mathord {+}}\mathfrak {k}_{2 k }\mathfrak {k}_{2 k +2}^{\mathord {+}}\mathfrak {k}_{2 k +3}-\textbf{T}^{\left( {\alpha }\right) }_{2k+1}\left( {z}\right) \mathfrak {k}_{2 k +2}^{\mathord {+}}\mathfrak {k}_{2 k +3}\\&=\zeta ^{ k +2}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) +\zeta \mathfrak {k}_{2 k }^{\mathord {+}}\mathfrak {k}_{2 k }+\zeta \textbf{T}^{\left( {\alpha }\right) }_{2k+1}\left( {z}\right) \\&\qquad -\mathfrak {k}_{2k+1}^{\mathord {+}}\mathfrak {k}_{2 k +2}-\textbf{T}^{\left( {\alpha }\right) }_{2k+1}\left( {z}\right) \mathfrak {k}_{2k+1}^{\mathord {+}}\mathfrak {k}_{2 k +2}-\mathfrak {k}_{2 k +2}^{\mathord {+}}\mathfrak {k}_{2 k +3}-\textbf{T}^{\left( {\alpha }\right) }_{2k+1}\left( {z}\right) \mathfrak {k}_{2 k +2}^{\mathord {+}}\mathfrak {k}_{2 k +3}\\&=\zeta ^{ k +2}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) +\zeta \mathfrak {k}_{2 k }^{\mathord {+}}\mathfrak {k}_{2 k }+\zeta \sum _{\ell =0}^{k-1}\zeta ^{k-\ell }\left( {\mathfrak {k}_{2\ell }^{\mathord {+}}\mathfrak {k}_{2\ell }-\mathfrak {k}_{2\ell +1}^{\mathord {+}}\mathfrak {k}_{2\ell +1}}\right) \\&\qquad -\mathfrak {k}_{2k+1}^{\mathord {+}}\mathfrak {k}_{2 k +2}-\sum _{\ell =0}^{k-1}\zeta ^{k-\ell }\left( {\mathfrak {k}_{2\ell }^{\mathord {+}}\mathfrak {k}_{2\ell }-\mathfrak {k}_{2\ell +1}^{\mathord {+}}\mathfrak {k}_{2\ell +1}}\right) \mathfrak {k}_{2k+1}^{\mathord {+}}\mathfrak {k}_{2 k +2}\\&\qquad -\mathfrak {k}_{2 k +2}^{\mathord {+}}\mathfrak {k}_{2 k +3}-\sum _{\ell =0}^{k-1}\zeta ^{k-\ell }\left( {\mathfrak {k}_{2\ell }^{\mathord {+}}\mathfrak {k}_{2\ell }-\mathfrak {k}_{2\ell +1}^{\mathord {+}}\mathfrak {k}_{2\ell +1}}\right) \mathfrak {k}_{2 k +2}^{\mathord {+}}\mathfrak {k}_{2 k +3}\\&=\zeta ^{ k +2}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) +\zeta \mathfrak {k}_{2 k }^{\mathord {+}}\mathfrak {k}_{2 k }+\sum _{\ell =0}^{k-1}\zeta ^{k+1-\ell } \\&\left( {\mathfrak {k}_{2\ell }^{\mathord {+}}\mathfrak {k}_{2\ell } -\mathfrak {k}_{2\ell +1}^{\mathord {+}}\mathfrak {k}_{2\ell +1}}\right) -\mathfrak {k}_{2k+1}^{\mathord {+}}\mathfrak {k}_{2 k +2}-\mathfrak {k}_{2 k +2}^{\mathord {+}}\mathfrak {k}_{2 k +3}. \end{aligned}\end{aligned}$$

(15.10)

From Notation 14.6, (15.6), and (15.9) we get

(15.11)

Let

$$\begin{aligned} \mathfrak {W}^{\left( {\alpha }\right) }_{2\left( { k +1}\right) +1}\left( {z}\right) \mathfrak {V}^{\left( {\alpha }\right) }_{2\left( { k +1}\right) +1}\left( {z}\right) = \begin{bmatrix} A &{} B \\ C &{} D \end{bmatrix} \end{aligned}$$

(15.12)

be the ${q\times q}$ block representation of $\mathfrak {W}^{\left( {\alpha }\right) }_{2\left( { k +1}\right) +1}\left( {z}\right) \mathfrak {V}^{\left( {\alpha }\right) }_{2\left( { k +1}\right) +1}\left( {z}\right)$. From (15.12) and (15.11) we obtain $A =\mathfrak {k}_{2\left( { k +1}\right) }\mathfrak {k}_{2\left( { k +1}\right) }^{\mathord {+}}$, whereas (15.12), (15.11), and (9.4) yield $C=\left( {I_{q}-\mathfrak {k}_{2k+1}^{\mathord {+}}\mathfrak {k}_{2k+1}}\right) \mathfrak {k}_{2 k +2}^{\mathord {+}}=O_{{q\times q}}$. Moreover, in view of (15.12), (15.11), (15.10), and (9.3), it follows

$$\begin{aligned}\begin{aligned} B&=-\left( {\zeta I_{q}-\mathfrak {k}_{2 k +2}\mathfrak {k}_{2k+1}^{\mathord {+}}-\mathfrak {k}_{2 k +3}\mathfrak {k}_{2 k +2}^{\mathord {+}}}\right) \mathfrak {k}_{2 k +2}+\mathfrak {k}_{2 k +2}R\\&=-\zeta \mathfrak {k}_{2 k +2}+\mathfrak {k}_{2 k +2}\mathfrak {k}_{2k+1}^{\mathord {+}}\mathfrak {k}_{2 k +2}+\mathfrak {k}_{2 k +3}\mathfrak {k}_{2 k +2}^{\mathord {+}}\mathfrak {k}_{2 k +2}\\&\qquad +\mathfrak {k}_{2 k +2}\biggl \{\zeta ^{ k +2}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) +\zeta \mathfrak {k}_{2 k }^{\mathord {+}}\mathfrak {k}_{2 k }\\&\qquad \qquad +\sum _{\ell =0}^{k-1}\zeta ^{k+1-\ell }\left( {\mathfrak {k}_{2\ell }^{\mathord {+}}\mathfrak {k}_{2\ell }-\mathfrak {k}_{2\ell +1}^{\mathord {+}}\mathfrak {k}_{2\ell +1}}\right) -\mathfrak {k}_{2k+1}^{\mathord {+}}\mathfrak {k}_{2 k +2}-\mathfrak {k}_{2 k +2}^{\mathord {+}}\mathfrak {k}_{2 k +3}\biggr \}\\&=-\zeta \mathfrak {k}_{2 k +2}+\mathfrak {k}_{2 k +2}\mathfrak {k}_{2k+1}^{\mathord {+}}\mathfrak {k}_{2 k +2}+\mathfrak {k}_{2 k +3}\mathfrak {k}_{2 k +2}^{\mathord {+}}\mathfrak {k}_{2 k +2}\\&\qquad +\zeta ^{ k +2}\mathfrak {k}_{2 k +2}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) +\zeta \mathfrak {k}_{2 k +2}\mathfrak {k}_{2 k }^{\mathord {+}}\mathfrak {k}_{2 k }\\&\qquad +\sum _{\ell =0}^{k-1}\zeta ^{k+1-\ell }\mathfrak {k}_{2 k +2}\left( {\mathfrak {k}_{2\ell }^{\mathord {+}}\mathfrak {k}_{2\ell }-\mathfrak {k}_{2\ell +1}^{\mathord {+}}\mathfrak {k}_{2\ell +1}}\right) \\&\qquad -\mathfrak {k}_{2 k +2}\mathfrak {k}_{2k+1}^{\mathord {+}}\mathfrak {k}_{2 k +2}-\mathfrak {k}_{2 k +2}\mathfrak {k}_{2 k +2}^{\mathord {+}}\mathfrak {k}_{2 k +3}\\&=-\zeta \mathfrak {k}_{2 k +2}+\mathfrak {k}_{2 k +2}\mathfrak {k}_{2k+1}^{\mathord {+}}\mathfrak {k}_{2 k +2}+\mathfrak {k}_{2 k +3}+\zeta \mathfrak {k}_{2 k +2}-\mathfrak {k}_{2 k +2}\mathfrak {k}_{2k+1}^{\mathord {+}}\mathfrak {k}_{2 k +2}-\mathfrak {k}_{2 k +3}\\&=O_{{q\times q}}. \end{aligned}\end{aligned}$$

Lastly, regarding (15.12), (15.11), and (15.10) and taking into account (9.3), (9.4), and (15.1), we conclude

$$\begin{aligned} D= & {} \mathfrak {k}_{2 k +2}^{\mathord {+}}\mathfrak {k}_{2 k +2}+\left( {I_{q}-\mathfrak {k}_{2k+1}^{\mathord {+}}\mathfrak {k}_{2k+1}}\right) R =\mathfrak {k}_{2 k +2}^{\mathord {+}}\mathfrak {k}_{2 k +2}+R-\mathfrak {k}_{2k+1}^{\mathord {+}}\mathfrak {k}_{2k+1}R\\= & {} \mathfrak {k}_{2 k +2}^{\mathord {+}}\mathfrak {k}_{2 k +2}+R-\mathfrak {k}_{2k+1}^{\mathord {+}}\mathfrak {k}_{2k+1}\biggl \{\zeta ^{ k +2}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) +\zeta \mathfrak {k}_{2 k }^{\mathord {+}}\mathfrak {k}_{2 k }\\{} & {} \qquad +\sum _{\ell =0}^{k-1}\zeta ^{k+1-\ell }\left( {\mathfrak {k}_{2\ell }^{\mathord {+}}\mathfrak {k}_{2\ell }-\mathfrak {k}_{2\ell +1}^{\mathord {+}}\mathfrak {k}_{2\ell +1}}\right) -\mathfrak {k}_{2k+1}^{\mathord {+}}\mathfrak {k}_{2 k +2}-\mathfrak {k}_{2 k +2}^{\mathord {+}}\mathfrak {k}_{2 k +3}\biggr \}\\{} & {} =\mathfrak {k}_{2 k +2}^{\mathord {+}}\mathfrak {k}_{2 k +2}+R-\zeta ^{ k +2}\mathfrak {k}_{2k+1}^{\mathord {+}}\mathfrak {k}_{2k+1}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) -\zeta \mathfrak {k}_{2k+1}^{\mathord {+}}\mathfrak {k}_{2k+1}\mathfrak {k}_{2 k }^{\mathord {+}}\mathfrak {k}_{2 k }\\{} & {} \qquad -\sum _{\ell =0}^{k-1}\zeta ^{k+1-\ell }\mathfrak {k}_{2k+1}^{\mathord {+}}\mathfrak {k}_{2k+1}\left( {\mathfrak {k}_{2\ell }^{\mathord {+}}\mathfrak {k}_{2\ell }-\mathfrak {k}_{2\ell +1}^{\mathord {+}}\mathfrak {k}_{2\ell +1}}\right) \\{} & {} \qquad +\mathfrak {k}_{2k+1}^{\mathord {+}}\mathfrak {k}_{2k+1}\mathfrak {k}_{2k+1}^{\mathord {+}}\mathfrak {k}_{2 k +2}+\mathfrak {k}_{2k+1}^{\mathord {+}}\mathfrak {k}_{2k+1}\mathfrak {k}_{2 k +2}^{\mathord {+}}\mathfrak {k}_{2 k +3}\\= & {} \mathfrak {k}_{2 k +2}^{\mathord {+}}\mathfrak {k}_{2 k +2}+R-\zeta \mathfrak {k}_{2k+1}^{\mathord {+}}\mathfrak {k}_{2k+1}+\mathfrak {k}_{2k+1}^{\mathord {+}}\mathfrak {k}_{2 k +2}+\mathfrak {k}_{2 k +2}^{\mathord {+}}\mathfrak {k}_{2 k +3}\\= & {} \mathfrak {k}_{2 k +2}^{\mathord {+}}\mathfrak {k}_{2 k +2}+\biggl \{\zeta ^{ k +2}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) +\zeta \mathfrak {k}_{2 k }^{\mathord {+}}\mathfrak {k}_{2 k }\\{} & {} \qquad +\sum _{\ell =0}^{k-1}\zeta ^{k+1-\ell }\left( {\mathfrak {k}_{2\ell }^{\mathord {+}}\mathfrak {k}_{2\ell }-\mathfrak {k}_{2\ell +1}^{\mathord {+}}\mathfrak {k}_{2\ell +1}}\right) -\mathfrak {k}_{2k+1}^{\mathord {+}}\mathfrak {k}_{2 k +2}-\mathfrak {k}_{2 k +2}^{\mathord {+}}\mathfrak {k}_{2 k +3}\biggr \}\\{} & {} \qquad -\zeta \mathfrak {k}_{2k+1}^{\mathord {+}}\mathfrak {k}_{2k+1}+\mathfrak {k}_{2k+1}^{\mathord {+}}\mathfrak {k}_{2 k +2}+\mathfrak {k}_{2 k +2}^{\mathord {+}}\mathfrak {k}_{2 k +3}\\= & {} \mathfrak {k}_{2 k +2}^{\mathord {+}}\mathfrak {k}_{2 k +2}+\zeta ^{ k +2}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) \\{} & {} \qquad +\zeta \left( {\mathfrak {k}_{2 k }^{\mathord {+}}\mathfrak {k}_{2 k }-\mathfrak {k}_{2k+1}^{\mathord {+}}\mathfrak {k}_{2k+1}}\right) +\sum _{\ell =0}^{k-1}\zeta ^{k+1-\ell }\left( {\mathfrak {k}_{2\ell }^{\mathord {+}}\mathfrak {k}_{2\ell }-\mathfrak {k}_{2\ell +1}^{\mathord {+}}\mathfrak {k}_{2\ell +1}}\right) \\= & {} \zeta ^{ k +2}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) +\mathfrak {k}_{2 k +2}^{\mathord {+}}\mathfrak {k}_{2 k +2}+\sum _{\ell =0}^{k}\zeta ^{\left( {k+1}\right) -\ell }\left( {\mathfrak {k}_{2\ell }^{\mathord {+}}\mathfrak {k}_{2\ell }-\mathfrak {k}_{2\ell +1}^{\mathord {+}}\mathfrak {k}_{2\ell +1}}\right) \\= & {} \zeta ^{k+2}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) +\mathfrak {k}_{2k+2}^{\mathord {+}}\mathfrak {k}_{2k+2}+\textbf{T}^{\left( {\alpha }\right) }_{2\left( {k+1}\right) +1}\left( {z}\right) . \end{aligned}$$

Hence, in view of (15.12), we have

$$\begin{aligned}{} & {} \mathfrak {W}^{\left( {\alpha }\right) }_{2\left( { k +1}\right) +1}\left( {z}\right) \mathfrak {V}^{\left( {\alpha }\right) }_{2\left( { k +1}\right) +1}\left( {z}\right) = \begin{bmatrix} A &{} B \\ C &{} D \end{bmatrix}\nonumber \\{} & {} \quad ={{\,\textrm{diag}\,}}\left( {\mathfrak {k}_{2k+2}\mathfrak {k}_{2k+2}^{\mathord {+}},\zeta ^{k+2}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) +\mathfrak {k}_{2k+2}^{\mathord {+}}\mathfrak {k}_{2k+2}+\textbf{T}^{\left( {\alpha }\right) }_{2\left( {k+1}\right) +1}\left( {z}\right) }\right) . \end{aligned}$$

Thus, (15.2) is checked for $n=k+1$ as well. Consequently, the assertion is inductively proved. $\square$

The proof of the following essential proposition is similar to that of Proposition 14.8, but differs in important details. In order to avoid a cumbersome technical argumentation, in both proofs the sequence of prescribed matrix moments is first extended to a sequence $(s_j)_{j=0}^{\infty }\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,\infty ,\alpha }$ and the sequence $(a_j)_{j=0}^{\infty }$ build according to Notation 7.1 is also considered. While in the proof of Proposition 14.8 we first used the $\mathbb {R}$-QMP $\left[ {(\mathfrak {a}_{k})_{k=0}^{\infty },(\mathfrak {b}_{k})_{k=0}^{\infty },(\mathfrak {c}_{k})_{k=0}^{\infty },(\mathfrak {d}_{k})_{k=0}^{\infty }} \right]$ and the first ${[\alpha ,\infty )}$-QMP $\left[ {(\textbf{a}_{k})_{k=0}^{\infty },(\textbf{b}_{k})_{k=0}^{\infty },(\textbf{c}_{k})_{k=0}^{\infty },(\textbf{d}_{k})_{k=0}^{\infty }} \right]$ associated with $(s_j)_{j=0}^{\infty }$ and after that argued with the help of the $\mathbb {R}$-QMP $\left[ {(\mathfrak {a}_{{\alpha ,k}})_{k=0}^{\infty },(\mathfrak {b}_{{\alpha ,k}})_{k=0}^{\infty },(\mathfrak {c}_{{\alpha ,k}})_{k=0}^{\infty },(\mathfrak {d}_{{\alpha ,k}})_{k=0}^{\infty }} \right]$ associated with $(a_j)_{j=0}^{\infty }$ and the second ${[\alpha ,\infty )}$-QMP $\left[ {(\textbf{a}_{{{\mathord {\circ }},k}})_{k=0}^{\infty },(\textbf{b}_{{{\mathord {\circ }},k}})_{k=0}^{\infty },(\textbf{c}_{{{\mathord {\circ }},k}})_{k=0}^{\infty },(\textbf{d}_{{{\mathord {\circ }},k}})_{k=0}^{\infty }} \right]$ associated with $(s_j)_{j=0}^{\infty }$, the situation in the proof of the following proposition is different. First we use the $\mathbb {R}$-QMP $\left[ {(\mathfrak {a}_{{\alpha ,k}})_{k=0}^{\infty },(\mathfrak {b}_{{\alpha ,k}})_{k=0}^{\infty },(\mathfrak {c}_{{\alpha ,k}})_{k=0}^{\infty },(\mathfrak {d}_{{\alpha ,k}})_{k=0}^{\infty }} \right]$ associated with $(a_j)_{j=0}^{\infty }$ and the second ${[\alpha ,\infty )}$-QMP $\left[ {(\textbf{a}_{{{\mathord {\circ }},k}})_{k=0}^{\infty },(\textbf{b}_{{{\mathord {\circ }},k}})_{k=0}^{\infty },(\textbf{c}_{{{\mathord {\circ }},k}})_{k=0}^{\infty },(\textbf{d}_{{{\mathord {\circ }},k}})_{k=0}^{\infty }} \right]$ associated with $(s_j)_{j=0}^{\infty }$ and after that we argue with the help of the $\mathbb {R}$-QMP $\left[ {(\mathfrak {a}_{k})_{k=0}^{\infty },(\mathfrak {b}_{k})_{k=0}^{\infty },(\mathfrak {c}_{k})_{k=0}^{\infty },(\mathfrak {d}_{k})_{k=0}^{\infty }} \right]$ and the first ${[\alpha ,\infty )}$-QMP $\left[ {(\textbf{a}_{k})_{k=0}^{\infty },(\textbf{b}_{k})_{k=0}^{\infty },(\textbf{c}_{k})_{k=0}^{\infty },(\textbf{d}_{k})_{k=0}^{\infty }} \right]$ associated with $(s_j)_{j=0}^{\infty }$.

Proposition 15.4

Let $n\in \mathbb {N}_0$, let $(s_j)_{j=0}^{2n+1}\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,2n+1,\alpha }$, let $w\in \Pi _{\mathord {+}}$, and let $X\in \mathscr {K}_{2n}\left( {w}\right) \cap \mathscr {K}_{{{\mathord {\circ }},2n}}\left( {w}\right)$. Then there exists a function $F\in {\mathcal {S}_{0,q}\left[ {\mathbb {C}\backslash {[\alpha ,\infty )};(s_j)_{j=0}^{2n+1},\preccurlyeq } \right] }$ fulfilling $F\left( {w}\right) =X$.

Proof

According to Corollary 7.4 there exists a sequence $(s_{j})_{j=2n+2}^{\infty }$ of complex ${q\times q}$ matrices such that $(s_j)_{j=0}^{\infty }\in \mathcal {K}^\succcurlyeq _{q,\infty ,\alpha }=\mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,\infty ,\alpha }$. From [10, Prop. 5.1] we know that $(a_j)_{j=0}^{\infty }\in \mathcal {K}^\succcurlyeq _{q,\infty ,\alpha }=\mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,\infty ,\alpha }$. Consequently, from Remark 7.6 we can infer $(s_j)_{j=0}^{\infty }\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,\infty }=\mathcal {H}^\succcurlyeq _{q,\infty }$ and $(a_j)_{j=0}^{\infty }\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,\infty }=\mathcal {H}^\succcurlyeq _{q,\infty }$. Let $(\mathfrak {k}_{j})_{j=0}^{\infty }$ be the $\mathcal {K}_\alpha$-parameter sequence of $(s_j)_{j=0}^{\infty }$, let $(\mathfrak {h}_{j})_{j=0}^{\infty }$ be the $\mathcal {H}$-parameter sequence of $(s_j)_{j=0}^{\infty }$ and let $(\mathfrak {h}_{{\alpha ,j}})_{j=0}^{\infty }$ be the $\mathcal {H}$-parameter sequence of $(a_j)_{j=0}^{\infty }$. According to Remark 9.2, we have then (14.19). Remark 9.3 shows (14.20). Since $w\in \Pi _{\mathord {+}}$, we have $\Im w\in (0,\infty )$. Recall that $\mathbb {K}_{{q\times q}}$ stands for the set of all contractive complex ${q\times q}$ matrices. For the sake of improved readability, from hereon our proof is divided into six parts.

Part 1: Regarding $X\in \mathscr {K}_{2n}\left( {w}\right) \cap \mathscr {K}_{{{\mathord {\circ }},2n}}\left( {w}\right)$, we have, in particular, $X\in \mathscr {K}_{{{\mathord {\circ }},2n}}\left( {w}\right)$. Hence, according to (13.3) and Notation 6.7, there exists a matrix $C_{\mathord {\circ }}\in \mathbb {K}_{{q\times q}}$ fulfilling

$$\begin{aligned} \eta X+s_{0} =\mathscr {C}_{{\alpha ,2n}}\left( {w}\right) +\left( {w-\overline{w}}\right) ^{-1}\mathscr {A}_{{\alpha ,2n}}\left( {w}\right) C_{\mathord {\circ }}\mathscr {B}_{{\alpha ,2n}}\left( {w}\right) , \end{aligned}$$

(15.13)

where $\eta :=w-\alpha$. Regarding $(a_j)_{j=0}^{\infty }\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,\infty }$, let $(\chi _{{\alpha ,j}})_{j=-1}^{\infty }$ be the sequence of $\chi$-functions associated with $(a_j)_{j=0}^{\infty }$. Let $E_{\mathord {\circ }}:=-\left[ {\chi _{{\alpha ,2n+1}}\left( {w}\right) } \right] ^*$, let $B_{\mathord {\circ }}:=\left( {\Im w}\right) ^{-1}\Im E_{\mathord {\circ }}$, let $P_{\mathord {\circ }}:=\mathbb {P}_{\mathcal {R}\left( {E_{\mathord {\circ }}}\right) }$ and $Q_{\mathord {\circ }}:=\mathbb {P}_{\mathcal {N}\left( {E_{\mathord {\circ }}}\right) }$. Then

$$\begin{aligned} \chi _{{\alpha ,2n+1}}\left( {w}\right) =-E_{\mathord {\circ }}^*,\; \left[ {\chi _{{\alpha ,2n+1}}\left( {w}\right) } \right] ^*=-E_{\mathord {\circ }}, \text { and } \left( {\Im w}\right) ^{-1}\Im \chi _{{\alpha ,2n+1}}\left( {w}\right) =B_{\mathord {\circ }}. \nonumber \\ \end{aligned}$$

(15.14)

From Remark 6.3 we can thus infer $B_{\mathord {\circ }}\in \mathbb {C}_\succcurlyeq ^{{q\times q}}$. Obviously, the (constant) matrix-valued functions $\phi _{\mathord {\circ }},\psi _{\mathord {\circ }}:\Pi _{\mathord {+}}\rightarrow \mathbb {C}^{{q\times q}}$ defined by

$$\begin{aligned} \phi _{\mathord {\circ }}\left( {z}\right) :=E_{\mathord {\circ }}\sqrt{B_{\mathord {\circ }}}^{\mathord {+}}-E_{\mathord {\circ }}^*\sqrt{B_{\mathord {\circ }}}^{\mathord {+}}C_{\mathord {\circ }}P_{\mathord {\circ }}\text { and } \psi _{\mathord {\circ }}\left( {z}\right) :=\sqrt{B_{\mathord {\circ }}}^{\mathord {+}}-\sqrt{B_{\mathord {\circ }}}^{\mathord {+}}C_{\mathord {\circ }}P_{\mathord {\circ }}+Q_{\mathord {\circ }}, \nonumber \\ \end{aligned}$$

(15.15)

respectively, fulfill $\mathbb {H}\left( {\phi _{\mathord {\circ }}}\right) =\mathbb {H}\left( {\psi _{\mathord {\circ }}}\right) =\Pi _{\mathord {+}}$. Proposition 6.4 yields $\mathcal {R}\left( {\Im \chi _{{\alpha ,2n+1}}\left( {z}\right) }\right) =\mathcal {R}\left( {\left[ {\chi _{{\alpha ,2n+1}}\left( {z}\right) } \right] ^*}\right) =\mathcal {R}\left( {\mathfrak {h}_{{\alpha ,2n}}}\right)$ and $\mathcal {N}\left( {\Im \chi _{{\alpha ,2n+1}}\left( {z}\right) }\right) =\mathcal {N}\left( {\left[ {\chi _{{\alpha ,2n+1}}\left( {z}\right) } \right] ^*}\right) =\mathcal {N}\left( {\mathfrak {h}_{{\alpha ,2n}}}\right)$ for all $z\in \Pi _{\mathord {+}}$. By virtue of (15.14), consequently,

$$\begin{aligned} \mathcal {R}\left( {B_{\mathord {\circ }}}\right)&=\mathcal {R}\left( {E_{\mathord {\circ }}}\right) =\mathcal {R}\left( {\mathfrak {h}_{{\alpha ,2n}}}\right){} & {} \text {and}&\mathcal {N}\left( {B_{\mathord {\circ }}}\right)&=\mathcal {N}\left( {E_{\mathord {\circ }}}\right) =\mathcal {N}\left( {\mathfrak {h}_{{\alpha ,2n}}}\right) \end{aligned}$$

(15.16)

follow. Regarding $\Im w\in (0,\infty )$ and taking into account $B_{\mathord {\circ }}\in \mathbb {C}_\succcurlyeq ^{{q\times q}}$ and $\mathcal {R}\left( {B_{\mathord {\circ }}}\right) =\mathcal {R}\left( {E_{\mathord {\circ }}}\right)$ as well as (15.15) and $C_{\mathord {\circ }}\in \mathbb {K}_{{q\times q}}$, we can apply Lemma A.16 to discern that

$$\begin{aligned} {{\,\textrm{rank}\,}}\begin{bmatrix}\phi _{\mathord {\circ }}\left( {z}\right) \\ \psi _{\mathord {\circ }}\left( {z}\right) \end{bmatrix}&=q{} & {} \text {and}{} & {} \Im \left( {\left[ {\psi _{\mathord {\circ }}\left( {z}\right) } \right] ^*\phi _{\mathord {\circ }}\left( {z}\right) }\right) \in \mathbb {C}_\succcurlyeq ^{{q\times q}}, \end{aligned}$$

(15.17)

that

$$\begin{aligned} P_{\mathord {\circ }}\phi _{\mathord {\circ }}\left( {z}\right) =\phi _{\mathord {\circ }}\left( {z}\right) ,\; \phi _{\mathord {\circ }}\left( {z}\right) P_{\mathord {\circ }}=\phi _{\mathord {\circ }}\left( {z}\right) , \text { and } \psi _{\mathord {\circ }}\left( {z}\right) P_{\mathord {\circ }}=\psi _{\mathord {\circ }}\left( {z}\right) -Q_{\mathord {\circ }}, \end{aligned}$$

(15.18)

and that

$$\begin{aligned} \sqrt{B_{\mathord {\circ }}}^{\mathord {+}}\left[ {\phi _{\mathord {\circ }}\left( {z}\right) -E_{\mathord {\circ }}\psi \left( {z}\right) } \right] \left[ {\phi _{\mathord {\circ }}\left( {z}\right) -E_{\mathord {\circ }}^*\psi _{\mathord {\circ }}\left( {z}\right) } \right] ^{\mathord {+}}\sqrt{B_{\mathord {\circ }}} =P_{\mathord {\circ }}C_{\mathord {\circ }}P_{\mathord {\circ }}\end{aligned}$$

(15.19)

hold true for all $z\in \Pi _{\mathord {+}}$. In view of Definition 4.2 and Remark 4.3, from $\mathbb {H}\left( {\phi _{\mathord {\circ }}}\right) =\mathbb {H}\left( {\psi _{\mathord {\circ }}}\right) =\Pi _{\mathord {+}}$ and (15.17) we recognize that the pair $\left( {\phi _{\mathord {\circ }}};{\psi _{\mathord {\circ }}}\right)$ belongs to $\mathcal{P}\mathcal{R}_{q}\left( {\Pi _{\mathord {+}}}\right)$ and that the set $\mathcal {D}_{\mathord {\circ }}:=\emptyset$ belongs to $\mathscr {D}\left( {\phi _{\mathord {\circ }},\psi _{\mathord {\circ }}}\right)$. Regarding (15.16), we have

$$\begin{aligned} P_{\mathord {\circ }}&=\mathbb {P}_{\mathcal {R}\left( {E_{\mathord {\circ }}}\right) }=\mathbb {P}_{\mathcal {R}\left( {\mathfrak {h}_{{\alpha ,2n}}}\right) }{} & {} \text {and}{} & {} Q_{\mathord {\circ }}=\mathbb {P}_{\mathcal {N}\left( {E_{\mathord {\circ }}}\right) }=\mathbb {P}_{\mathcal {N}\left( {\mathfrak {h}_{{\alpha ,2n}}}\right) }. \end{aligned}$$

(15.20)

From (15.20) and (15.18) we receive $\mathbb {P}_{\mathcal {R}\left( {\mathfrak {h}_{{\alpha ,2n}}}\right) }\phi _{\mathord {\circ }}\left( {z}\right) =\phi _{\mathord {\circ }}\left( {z}\right)$ for all $z\in \Pi _{\mathord {+}}$. According to Notation 4.5, hence $\left( {\phi _{\mathord {\circ }}};{\psi _{\mathord {\circ }}}\right) \in \mathcal {P}\left[ {\mathfrak {h}_{{\alpha ,2n}}} \right]$. Let $\left[ {(\mathfrak {a}_{{\alpha ,k}})_{k=0}^{\infty },(\mathfrak {b}_{{\alpha ,k}})_{k=0}^{\infty },(\mathfrak {c}_{{\alpha ,k}})_{k=0}^{\infty },(\mathfrak {d}_{{\alpha ,k}})_{k=0}^{\infty }} \right]$ be the $\mathbb {R}$-QMP associated with $(a_j)_{j=0}^{\infty }$ and let $\left[ {(\textbf{a}_{{{\mathord {\circ }},k}})_{k=0}^{\infty },(\textbf{b}_{{{\mathord {\circ }},k}})_{k=0}^{\infty },(\textbf{c}_{{{\mathord {\circ }},k}})_{k=0}^{\infty },(\textbf{d}_{{{\mathord {\circ }},k}})_{k=0}^{\infty }} \right]$ be the second ${[\alpha ,\infty )}$-QMP associated with $(s_j)_{j=0}^{\infty }$. Regarding $\mathcal {D}_{\mathord {\circ }}\in \mathscr {D}\left( {\phi _{\mathord {\circ }},\psi _{\mathord {\circ }}}\right)$ as well as $\mathcal {D}_{\mathord {\circ }}=\emptyset$, we can infer from Corollary 12.17 for all $z\in \Pi _{\mathord {+}}$ then

$$\begin{aligned} \det \left( {\mathfrak {b}_{{\alpha ,n}}\left( {z}\right) \mathfrak {h}_{{\alpha ,2n}}^{\mathord {+}}\phi _{\mathord {\circ }}\left( {z}\right) +\mathfrak {b}_{{\alpha ,n+1}}\left( {z}\right) \psi _{\mathord {\circ }}\left( {z}\right) }\right)&\ne 0, \end{aligned}$$

(15.21)

$$\begin{aligned} \det \left( {\textbf{b}_{{{\mathord {\circ }},n}}\left( {z}\right) \mathfrak {h}_{{\alpha ,2n}}^{\mathord {+}}\phi _{\mathord {\circ }}\left( {z}\right) +\textbf{b}_{{{\mathord {\circ }},n+1}}\left( {z}\right) \psi _{\mathord {\circ }}\left( {z}\right) }\right)&\ne 0, \end{aligned}$$

(15.22)

and

$$\begin{aligned}{} & {} -\left[ {\mathfrak {a}_{{\alpha ,n}}\left( {z}\right) \mathfrak {h}_{{\alpha ,2n}}^{\mathord {+}}\phi _{\mathord {\circ }}\left( {z}\right) +\mathfrak {a}_{{\alpha ,n+1}}\left( {z}\right) \psi _{\mathord {\circ }}\left( {z}\right) } \right] \nonumber \\{} & {} \quad \times \left[ {\mathfrak {b}_{{\alpha ,n}}\left( {z}\right) \mathfrak {h}_{{\alpha ,2n}}^{\mathord {+}}\phi _{\mathord {\circ }}\left( {z}\right) +\mathfrak {b}_{{\alpha ,n+1}}\left( {z}\right) \psi _{\mathord {\circ }}\left( {z}\right) } \right] ^{-1}\nonumber \\{} & {} \quad =-\left[ {\textbf{a}_{{{\mathord {\circ }},n}}\left( {z}\right) \mathfrak {h}_{{\alpha ,2n}}^{\mathord {+}}\phi _{\mathord {\circ }}\left( {z}\right) +\textbf{a}_{{{\mathord {\circ }},n+1}}\left( {z}\right) \psi _{\mathord {\circ }}\left( {z}\right) } \right] \nonumber \\{} & {} \quad \times \left[ {\textbf{b}_{{{\mathord {\circ }},n}}\left( {z}\right) \mathfrak {h}_{{\alpha ,2n}}^{\mathord {+}}\phi _{\mathord {\circ }}\left( {z}\right) +\textbf{b}_{{{\mathord {\circ }},n+1}}\left( {z}\right) \psi _{\mathord {\circ }}\left( {z}\right) } \right] ^{-1}. \end{aligned}$$

(15.23)

Setting

$$\begin{aligned} S_{\mathord {\circ }}&:=\phi _{\mathord {\circ }}\left( {w}\right){} & {} \text {and}&T_{\mathord {\circ }}&:=\psi _{\mathord {\circ }}\left( {w}\right) \end{aligned}$$

(15.24)

we see, in view of (15.21), (15.22), and (15.23), that

$$\begin{aligned} {\mathfrak {R}}_{\mathord {\circ }}:=\mathfrak {b}_{{\alpha ,n}}\left( {w}\right) \mathfrak {h}_{{\alpha ,2n}}^{\mathord {+}}S_{\mathord {\circ }}+\mathfrak {b}_{{\alpha ,n+1}}\left( {w}\right) T_{\mathord {\circ }}\,\text {and}\, {\textbf{R}}_{\mathord {\circ }}:=\textbf{b}_{{{\mathord {\circ }},n}}\left( {w}\right) \mathfrak {h}_{{\alpha ,2n}}^{\mathord {+}}S_{\mathord {\circ }}+\textbf{b}_{{{\mathord {\circ }},n+1}}\left( {w}\right) T_{\mathord {\circ }}\nonumber \\ \end{aligned}$$

(15.25)

satisfy $\det {\mathfrak {R}}_{\mathord {\circ }}\ne 0$ and $\det {\textbf{R}}_{\mathord {\circ }}\ne 0$ as well as

$$\begin{aligned}{} & {} -\left[ {\mathfrak {a}_{{\alpha ,n}}\left( {w}\right) \mathfrak {h}_{{\alpha ,2n}}^{\mathord {+}}S_{\mathord {\circ }}+\mathfrak {a}_{{\alpha ,n+1}}\left( {w}\right) T_{\mathord {\circ }}} \right] {\mathfrak {R}}_{\mathord {\circ }}^{-1}\nonumber \\{} & {} \quad =-\left[ {\textbf{a}_{{{\mathord {\circ }},n}}\left( {w}\right) \mathfrak {h}_{{\alpha ,2n}}^{\mathord {+}}S_{\mathord {\circ }}+\textbf{a}_{{{\mathord {\circ }},n+1}}\left( {w}\right) T_{\mathord {\circ }}} \right] {\textbf{R}}_{\mathord {\circ }}^{-1}. \end{aligned}$$

(15.26)

Now we are going to justify that all assumptions for the application of Proposition 6.11 to the sequence $(a_j)_{j=0}^{2n+1}$ are satisfied. Because of $(a_j)_{j=0}^{\infty }\in \mathcal {H}^\succcurlyeq _{q,\infty }$, we have $(a_j)_{j=0}^{2n+1}\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,2n+1}$. According to (15.20), we have $P_{\mathord {\circ }}=\mathbb {P}_{\mathcal {R}\left( {\mathfrak {h}_{{\alpha ,2n}}}\right) }$ and $Q_{\mathord {\circ }}=\mathbb {P}_{\mathcal {N}\left( {\mathfrak {h}_{{\alpha ,2n}}}\right) }$. In view of $\Im w\in (0,\infty )$ and (15.24), from (15.17) and (15.18), we see $\left( {\Im w}\right) \Im \left( {T_{\mathord {\circ }}^*S_{\mathord {\circ }}}\right) \in \mathbb {C}_\succcurlyeq ^{{q\times q}}$ as well as

$$\begin{aligned} P_{\mathord {\circ }}S_{\mathord {\circ }}=S_{\mathord {\circ }},\quad S_{\mathord {\circ }}P_{\mathord {\circ }}=S_{\mathord {\circ }},\quad \quad \text {and}\quad T_{\mathord {\circ }}P_{\mathord {\circ }}=T_{\mathord {\circ }}-Q_{\mathord {\circ }}. \end{aligned}$$

(15.27)

In view of (15.25) and $\det {\mathfrak {R}}_{\mathord {\circ }}\ne 0$, Proposition 6.11 shows then that the matrix

$$\begin{aligned}{} & {} {\textbf{K}}_{\mathord {\circ }}:=\sqrt{\left( {\Im w}\right) ^{-1}\Im \chi _{{\alpha ,2n+1}}\left( {w}\right) }^{\mathord {+}}\left( {S_{\mathord {\circ }}+\left[ {\chi _{{\alpha ,2n+1}}\left( {w}\right) } \right] ^*T_{\mathord {\circ }}}\right) \nonumber \\{} & {} \quad \times \left( {S_{\mathord {\circ }}+\chi _{{\alpha ,2n+1}}\left( {w}\right) T_{\mathord {\circ }}}\right) ^{\mathord {+}}\sqrt{\left( {\Im w}\right) ^{-1}\Im \chi _{{\alpha ,2n+1}}\left( {w}\right) } \end{aligned}$$

(15.28)

is contractive and fulfills

$$\begin{aligned}{} & {} -\left[ {\mathfrak {a}_{{\alpha ,n}}\left( {w}\right) \mathfrak {h}_{{\alpha ,2n}}^{\mathord {+}}S_{\mathord {\circ }}+\mathfrak {a}_{{\alpha ,n+1}}\left( {w}\right) T_{\mathord {\circ }}} \right] {\mathfrak {R}}_{\mathord {\circ }}^{-1}\nonumber \\{} & {} \quad =\mathscr {C}_{{\alpha ,2n+1}}\left( {w}\right) +\left( {w-\overline{w}}\right) ^{-1}\mathscr {A}_{{\alpha ,2n+1}}\left( {w}\right) {\textbf{K}}_{\mathord {\circ }}\mathscr {B}_{{\alpha ,2n+1}}\left( {w}\right) . \end{aligned}$$

(15.29)

By virtue of (15.28), (15.14), (15.24), and (15.19), we discern

$$\begin{aligned} {\textbf{K}}_{\mathord {\circ }}=\sqrt{B_{\mathord {\circ }}}^{\mathord {+}}\left[ {\phi _{\mathord {\circ }}\left( {w}\right) -E_{\mathord {\circ }}\psi _{\mathord {\circ }}\left( {w}\right) } \right] \left[ {\phi _{\mathord {\circ }}\left( {w}\right) -E_{\mathord {\circ }}^*\psi _{\mathord {\circ }}\left( {w}\right) } \right] ^{\mathord {+}}\sqrt{B_{\mathord {\circ }}} =P_{\mathord {\circ }}C_{\mathord {\circ }}P. \end{aligned}$$

(15.30)

The application of Lemma 6.9 to the sequence $(a_j)_{j=0}^{\infty }\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,\infty }$ yields

$$\begin{aligned} \mathscr {A}_{{\alpha ,2n}}\left( {w}\right) =\mathscr {A}_{{\alpha ,2n+1}}\left( {w}\right) ,\; \mathscr {B}_{{\alpha ,2n}}\left( {w}\right) =\mathscr {B}_{{\alpha ,2n+1}}\left( {w}\right) ,\; \mathscr {C}_{{\alpha ,2n}}\left( {w}\right) =\mathscr {C}_{{\alpha ,2n+1}}\left( {w}\right) .\nonumber \\ \end{aligned}$$

(15.31)

Regarding $(a_j)_{j=0}^{\infty }\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,\infty }$, from Proposition 6.10 we get $\mathcal {N}\left( {\mathscr {A}_{{\alpha ,2n}}\left( {w}\right) }\right) =\mathcal {N}\left( {\mathfrak {h}_{{\alpha ,2n}}}\right)$ and $\mathcal {R}\left( {\mathscr {B}_{{\alpha ,2n}}\left( {w}\right) }\right) =\mathcal {R}\left( {\mathfrak {h}_{{\alpha ,2n}}}\right)$. According to (15.20), we have $P_{\mathord {\circ }}=\mathbb {P}_{\mathcal {R}\left( {\mathfrak {h}_{{\alpha ,2n}}}\right) }$ and $Q_{\mathord {\circ }}=\mathbb {P}_{\mathcal {N}\left( {\mathfrak {h}_{{\alpha ,2n}}}\right) }$. Clearly, then $P_{\mathord {\circ }}\mathscr {B}_{{\alpha ,2n}}\left( {w}\right) =\mathscr {B}_{{\alpha ,2n}}\left( {w}\right)$ follows. By virtue of Remark A.8, we see furthermore $\mathcal {R}\left( {P_{\mathord {\circ }}}\right) =\mathcal {R}\left( {\mathfrak {h}_{{\alpha ,2n}}}\right)$ and $\mathcal {R}\left( {Q_{\mathord {\circ }}}\right) =\mathcal {N}\left( {\mathfrak {h}_{{\alpha ,2n}}}\right)$. Hence, we get $\mathscr {A}_{{\alpha ,2n}}\left( {w}\right) Q_{\mathord {\circ }}=O_{{q\times q}}$. Remark A.7 provides $\mathcal {N}\left( {\mathfrak {h}_{{\alpha ,2n}}^*}\right) =\mathcal {R}\left( {\mathfrak {h}_{{\alpha ,2n}}}\right) ^\bot$. Since, in view of $(a_j)_{j=0}^{\infty }\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,\infty }$, Remark 5.3 yields $\mathfrak {h}_{{\alpha ,2n}}^*=\mathfrak {h}_{{\alpha ,2n}}$, then $\mathcal {N}\left( {\mathfrak {h}_{{\alpha ,2n}}}\right) =\mathcal {R}\left( {\mathfrak {h}_{{\alpha ,2n}}}\right) ^\bot$ follows. Thus, Remark A.9 yields $P_{\mathord {\circ }}+Q_{\mathord {\circ }}=I_{q}$, implying $\mathscr {A}_{{\alpha ,2n}}\left( {w}\right) P_{\mathord {\circ }}=\mathscr {A}_{{\alpha ,2n}}\left( {w}\right)$. Consequently, we infer $\mathscr {A}_{{\alpha ,2n}}\left( {w}\right) P_{\mathord {\circ }}C_{\mathord {\circ }}P_{\mathord {\circ }}\mathscr {B}_{{\alpha ,2n}}\left( {w}\right) =\mathscr {A}_{{\alpha ,2n}}\left( {w}\right) C_{\mathord {\circ }}\mathscr {B}_{{\alpha ,2n}}\left( {w}\right)$. Using additionally (15.13), (15.31), (15.30), (15.29), and (15.26), we conclude

$$\begin{aligned} \eta X+s_{0}= & {} \mathscr {C}_{{\alpha ,2n}}\left( {w}\right) +\left( {w-\overline{w}}\right) ^{-1}\mathscr {A}_{{\alpha ,2n}}\left( {w}\right) C_{\mathord {\circ }}\mathscr {B}_{{\alpha ,2n}}\left( {w}\right) \nonumber \\= & {} \mathscr {C}_{{\alpha ,2n}}\left( {w}\right) +\left( {w-\overline{w}}\right) ^{-1}\mathscr {A}_{{\alpha ,2n}}\left( {w}\right) P_{\mathord {\circ }}C_{\mathord {\circ }}P_{\mathord {\circ }}\mathscr {B}_{{\alpha ,2n}}\left( {w}\right) \nonumber \\= & {} \mathscr {C}_{{\alpha ,2n+1}}\left( {w}\right) +\left( {w-\overline{w}}\right) ^{-1}\mathscr {A}_{{\alpha ,2n+1}}\left( {w}\right) {\textbf{K}}_{\mathord {\circ }}\mathscr {B}_{{\alpha ,2n+1}}\left( {w}\right) \nonumber \\= & {} -\left[ {\mathfrak {a}_{{\alpha ,n}}\left( {w}\right) \mathfrak {h}_{{\alpha ,2n}}^{\mathord {+}}S_{\mathord {\circ }}+\mathfrak {a}_{{\alpha ,n+1}}\left( {w}\right) T_{\mathord {\circ }}} \right] {\mathfrak {R}}_{\mathord {\circ }}^{-1}\nonumber \\= & {} -\left[ {\textbf{a}_{{{\mathord {\circ }},n}}\left( {w}\right) \mathfrak {h}_{{\alpha ,2n}}^{\mathord {+}}S_{\mathord {\circ }}+\textbf{a}_{{{\mathord {\circ }},n+1}}\left( {w}\right) T_{\mathord {\circ }}} \right] {\textbf{R}}_{\mathord {\circ }}^{-1}. \end{aligned}$$

Using additionally $\det {\textbf{R}}_{\mathord {\circ }}\ne 0$, (15.25), (14.19), and Lemma 14.2(b), we then obtain

$$\begin{aligned} \begin{aligned} \begin{bmatrix} \eta X+s_{0}\\ I_{q}\end{bmatrix}&= \begin{bmatrix}-\left[ {\textbf{a}_{{{\mathord {\circ }},n}}\left( {w}\right) \mathfrak {h}_{{\alpha ,2n}}^{\mathord {+}}S_{\mathord {\circ }}+\textbf{a}_{{{\mathord {\circ }},n+1}}\left( {w}\right) T_{\mathord {\circ }}} \right] {\textbf{R}}_{\mathord {\circ }}^{-1}\\ {\textbf{R}}_{\mathord {\circ }}{\textbf{R}}_{\mathord {\circ }}^{-1}\end{bmatrix}\\&= \begin{bmatrix} -\textbf{a}_{{{\mathord {\circ }},n}}\left( {w}\right) \mathfrak {h}_{{\alpha ,2n}}^{\mathord {+}}S_{\mathord {\circ }}-\textbf{a}_{{{\mathord {\circ }},n+1}}\left( {w}\right) T_{\mathord {\circ }}\\ \textbf{b}_{{{\mathord {\circ }},n}}\left( {w}\right) \mathfrak {h}_{{\alpha ,2n}}^{\mathord {+}}S_{\mathord {\circ }}+\textbf{b}_{{{\mathord {\circ }},n+1}}\left( {w}\right) T_{\mathord {\circ }}\end{bmatrix} {\textbf{R}}_{\mathord {\circ }}^{-1}\\&= \begin{bmatrix} -\textbf{a}_{{{\mathord {\circ }},n}}\left( {w}\right) \mathfrak {k}_{2n+1}^{\mathord {+}}&{}-\textbf{a}_{{{\mathord {\circ }},n+1}}\left( {w}\right) \\ \textbf{b}_{{{\mathord {\circ }},n}}\left( {w}\right) \mathfrak {k}_{2n+1}^{\mathord {+}}&{}\textbf{b}_{{{\mathord {\circ }},n+1}}\left( {w}\right) \end{bmatrix} \begin{bmatrix} S_{\mathord {\circ }}\\ T_{\mathord {\circ }}\end{bmatrix} {\textbf{R}}_{\mathord {\circ }}^{-1}\\&=\mathfrak {V}^{\left( {\alpha }\right) }_{2n+2}\left( {w}\right) \begin{bmatrix} S_{\mathord {\circ }}\\ T_{\mathord {\circ }}\end{bmatrix} {\textbf{R}}_{\mathord {\circ }}^{-1}. \end{aligned} \end{aligned}$$

(15.32)

Part 2: Due to $X\in \mathscr {K}_{2n}\left( {w}\right) \cap \mathscr {K}_{{{\mathord {\circ }},2n}}\left( {w}\right)$, we have, in particular, $X\in \mathscr {K}_{2n}\left( {w}\right)$. Hence, according to (13.1) and Notation 6.7, there exists a matrix $C\in \mathbb {K}_{{q\times q}}$ fulfilling

$$\begin{aligned} X =\mathscr {C}_{2n}\left( {w}\right) +\left( {w-\overline{w}}\right) ^{-1}\mathscr {A}_{2n}\left( {w}\right) C\mathscr {B}_{2n}\left( {w}\right) . \end{aligned}$$

(15.33)

Regarding $(s_j)_{j=0}^{\infty }\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,\infty }$, let $(\chi _{j})_{j=-1}^{\infty }$ be the sequence of $\chi$-functions associated with $(s_j)_{j=0}^{\infty }$. Let $E:=-\left[ {\chi _{2n+1}\left( {w}\right) } \right] ^*$, let $B:=\left( {\Im w}\right) ^{-1}\Im E$, let $P:=\mathbb {P}_{\mathcal {R}\left( {E}\right) }$ and $Q:=\mathbb {P}_{\mathcal {N}\left( {E}\right) }$. Clearly, then

$$\begin{aligned} \chi _{2n+1}\left( {w}\right) =-E^*,\, \left[ {\chi _{2n+1}\left( {w}\right) } \right] ^*=-E, \,\text {and}\, \left( {\Im w}\right) ^{-1}\Im \chi _{2n+1}\left( {w}\right) =B.\nonumber \\ \end{aligned}$$

(15.34)

From Remark 6.3 we can thus infer $B\in \mathbb {C}_\succcurlyeq ^{{q\times q}}$. Obviously, the (constant) matrix-valued functions $\phi ,\psi :\Pi _{\mathord {+}}\rightarrow \mathbb {C}^{{q\times q}}$ defined by

$$\begin{aligned} \phi \left( {z}\right) :=E\sqrt{B}^{\mathord {+}}-E^*\sqrt{B}^{\mathord {+}}CP\text { and } \psi \left( {z}\right) :=\sqrt{B}^{\mathord {+}}-\sqrt{B}^{\mathord {+}}CP+Q, \end{aligned}$$

(15.35)

respectively, fulfill $\mathbb {H}\left( {\phi }\right) =\mathbb {H}\left( {\psi }\right) =\Pi _{\mathord {+}}$. Proposition 6.4 yields $\mathcal {R}\left( {\Im \chi _{2n+1}\left( {z}\right) }\right) =\mathcal {R}\left( {\left[ {\chi _{2n+1}\left( {z}\right) } \right] ^*}\right) =\mathcal {R}\left( {\mathfrak {h}_{2n}}\right)$ and $\mathcal {N}\left( {\Im \chi _{2n+1}\left( {z}\right) }\right) =\mathcal {N}\left( {\left[ {\chi _{2n+1}\left( {z}\right) } \right] ^*}\right) =\mathcal {N}\left( {\mathfrak {h}_{2n}}\right)$ for all $z\in \Pi _{\mathord {+}}$. By virtue of (15.34), consequently,

$$\begin{aligned} \mathcal {R}\left( {B}\right)&=\mathcal {R}\left( {E}\right) =\mathcal {R}\left( {\mathfrak {h}_{2n}}\right){} & {} \text {and}&\mathcal {N}\left( {B}\right)&=\mathcal {N}\left( {E}\right) =\mathcal {N}\left( {\mathfrak {h}_{2n}}\right) \end{aligned}$$

(15.36)

follow. Regarding $\Im w\in (0,\infty )$ and taking into account $B\in \mathbb {C}_\succcurlyeq ^{{q\times q}}$ and $\mathcal {R}\left( {B}\right) =\mathcal {R}\left( {E}\right)$ as well as (15.35) and $C\in \mathbb {K}_{{q\times q}}$, we can apply Lemma A.16 to discern that

$$\begin{aligned} {{\,\textrm{rank}\,}}\begin{bmatrix}\phi \left( {z}\right) \\ \psi \left( {z}\right) \end{bmatrix}=q \quad \text {and}\quad \Im \left( {\left[ {\psi \left( {z}\right) } \right] ^*\phi \left( {z}\right) }\right) \in \mathbb {C}_\succcurlyeq ^{{q\times q}}, \end{aligned}$$

(15.37)

that

$$\begin{aligned} P\phi \left( {z}\right)&=\phi \left( {z}\right) ,&\phi \left( {z}\right) P&=\phi \left( {z}\right) ,{} & {} \text {and}&\psi \left( {z}\right) P&=\psi \left( {z}\right) -Q, \end{aligned}$$

(15.38)

and that

$$\begin{aligned} \sqrt{B}^{\mathord {+}}\left[ {\phi \left( {z}\right) -E\psi \left( {z}\right) } \right] \left[ {\phi \left( {z}\right) -E^*\psi \left( {z}\right) } \right] ^{\mathord {+}}\sqrt{B} =PCP\end{aligned}$$

(15.39)

hold true for all $z\in \Pi _{\mathord {+}}$. In view of Definition 4.2 and Remark 4.3, from $\mathbb {H}\left( {\phi }\right) =\mathbb {H}\left( {\psi }\right) =\Pi _{\mathord {+}}$ and (15.37) we recognize that the pair $\left( {\phi };{\psi }\right)$ belongs to $\mathcal{P}\mathcal{R}_{q}\left( {\Pi _{\mathord {+}}}\right)$ and that the set $\mathcal {D}:=\emptyset$ belongs to $\mathscr {D}\left( {\phi ,\psi }\right)$. Regarding (15.36), we have

$$\begin{aligned} P&=\mathbb {P}_{\mathcal {R}\left( {E}\right) }=\mathbb {P}_{\mathcal {R}\left( {\mathfrak {h}_{2n}}\right) }{} & {} \text {and}&Q&=\mathbb {P}_{\mathcal {N}\left( {E}\right) }=\mathbb {P}_{\mathcal {N}\left( {\mathfrak {h}_{2n}}\right) }. \end{aligned}$$

(15.40)

From (15.40) and (15.38) we receive $\mathbb {P}_{\mathcal {R}\left( {\mathfrak {h}_{2n}}\right) }\phi \left( {z}\right) =\phi \left( {z}\right)$ for all $z\in \Pi _{\mathord {+}}$. According to Notation 4.5, hence $\left( {\phi };{\psi }\right) \in \mathcal {P}\left[ {\mathfrak {h}_{2n}} \right]$. Let $\left[ {(\mathfrak {a}_{k})_{k=0}^{\infty },(\mathfrak {b}_{k})_{k=0}^{\infty },(\mathfrak {c}_{k})_{k=0}^{\infty },(\mathfrak {d}_{k})_{k=0}^{\infty }} \right]$ be the $\mathbb {R}$-QMP associated with $(s_j)_{j=0}^{\infty }$ and let $\left[ {(\textbf{a}_{k})_{k=0}^{\infty },(\textbf{b}_{k})_{k=0}^{\infty },(\textbf{c}_{k})_{k=0}^{\infty },(\textbf{d}_{k})_{k=0}^{\infty }} \right]$ be the first ${[\alpha ,\infty )}$-QMP associated with $(s_j)_{j=0}^{\infty }$. Regarding $\mathcal {D}\in \mathscr {D}\left( {\phi ,\psi }\right)$ as well as $\mathcal {D}=\emptyset$, we can infer from Corollary 11.14 for all $z\in \Pi _{\mathord {+}}$ then

$$\begin{aligned} \det \left( {\mathfrak {b}_{n}\left( {z}\right) \mathfrak {h}_{2n}^{\mathord {+}}\phi \left( {z}\right) +\mathfrak {b}_{n+1}\left( {z}\right) \psi \left( {z}\right) }\right)&\ne 0,\end{aligned}$$

(15.41)

$$\begin{aligned} \det \left( {\textbf{b}_{n}\left( {z}\right) \mathfrak {h}_{2n}^{\mathord {+}}\phi \left( {z}\right) +\textbf{b}_{n+1}\left( {z}\right) \psi \left( {z}\right) }\right)&\ne 0, \end{aligned}$$

(15.42)

and

$$\begin{aligned}{} & {} -\left[ {\mathfrak {a}_{n}\left( {z}\right) \mathfrak {h}_{2n}^{\mathord {+}}\phi \left( {z}\right) +\mathfrak {a}_{n+1}\left( {z}\right) \psi \left( {z}\right) } \right] \left[ {\mathfrak {b}_{n}\left( {z}\right) \mathfrak {h}_{2n}^{\mathord {+}}\phi \left( {z}\right) +\mathfrak {b}_{n+1}\left( {z}\right) \psi \left( {z}\right) } \right] ^{-1}\nonumber \\{} & {} \quad =-\left[ {\textbf{a}_{n}\left( {z}\right) \mathfrak {h}_{2n}^{\mathord {+}}\phi \left( {z}\right) +\textbf{a}_{n+1}\left( {z}\right) \psi \left( {z}\right) } \right] \left[ {\textbf{b}_{n}\left( {z}\right) \mathfrak {h}_{2n}^{\mathord {+}}\phi \left( {z}\right) +\textbf{b}_{n+1}\left( {z}\right) \psi \left( {z}\right) } \right] ^{-1}.\nonumber \\ \end{aligned}$$

(15.43)

Setting

$$\begin{aligned} S:=\phi \left( {w}\right) \quad \text {and}\quad T:=\psi \left( {w}\right) \end{aligned}$$

(15.44)

we see, in view of (15.41), (15.42), and (15.43), that

$$\begin{aligned} {\mathfrak {R}}&:=\mathfrak {b}_{n}\left( {w}\right) \mathfrak {h}_{2n}^{\mathord {+}}S+\mathfrak {b}_{n+1}\left( {w}\right) T{} & {} \text {and}&{\textbf{R}}&:=\textbf{b}_{n}\left( {w}\right) \mathfrak {h}_{2n}^{\mathord {+}}S+\textbf{b}_{n+1}\left( {w}\right) T\end{aligned}$$

(15.45)

satisfy $\det {\mathfrak {R}}\ne 0$ and $\det {\textbf{R}}\ne 0$ as well as

$$\begin{aligned} -\left[ {\mathfrak {a}_{n}\left( {w}\right) \mathfrak {h}_{2n}^{\mathord {+}}S+\mathfrak {a}_{n+1}\left( {w}\right) T} \right] {\mathfrak {R}}^{-1}=-\left[ {\textbf{a}_{n}\left( {w}\right) \mathfrak {h}_{2n}^{\mathord {+}}S+\textbf{a}_{n+1}\left( {w}\right) T} \right] {\textbf{R}}^{-1}. \end{aligned}$$

(15.46)

Now we are going to justify that all assumptions for the application of Proposition 6.11 to the sequence $(s_j)_{j=0}^{2n+1}$ are satisfied. Because of $(s_j)_{j=0}^{\infty }\in \mathcal {H}^\succcurlyeq _{q,\infty }$, we have $(s_j)_{j=0}^{2n+1}\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,2n+1}$. According to (15.40), we have $P=\mathbb {P}_{\mathcal {R}\left( {\mathfrak {h}_{2n}}\right) }$ and $Q=\mathbb {P}_{\mathcal {N}\left( {\mathfrak {h}_{2n}}\right) }$. In view of $\Im w\in (0,\infty )$ and (15.44), from (15.37) and (15.38), we see $\left( {\Im w}\right) \Im \left( {T^*S}\right) \in \mathbb {C}_\succcurlyeq ^{{q\times q}}$ as well as

$$\begin{aligned} PS=S,\quad SP=S,\quad \text {and}\quad TP=T-Q. \end{aligned}$$

(15.47)

Using additionally (15.45) and $\det {\mathfrak {R}}\ne 0$, the application of Proposition 6.11 yields that the matrix

$$\begin{aligned}{} & {} {\textbf{K}} :=\sqrt{\left( {\Im w}\right) ^{-1}\Im \chi _{2n+1}\left( {w}\right) }^{\mathord {+}}\left( {S+\left[ {\chi _{2n+1}\left( {w}\right) } \right] ^*T}\right) \nonumber \\{} & {} \quad \times \left( {S+\chi _{2n+1}\left( {w}\right) T}\right) ^{\mathord {+}}\sqrt{\left( {\Im w}\right) ^{-1}\Im \chi _{2n+1}\left( {w}\right) } \end{aligned}$$

(15.48)

is contractive and fulfills

$$\begin{aligned}{} & {} -\left[ {\mathfrak {a}_{n}\left( {w}\right) \mathfrak {h}_{2n}^{\mathord {+}}S+\mathfrak {a}_{n+1}\left( {w}\right) T} \right] {\mathfrak {R}}^{-1}\nonumber \\{} & {} \quad =\mathscr {C}_{2n+1}\left( {w}\right) +\left( {w-\overline{w}}\right) ^{-1}\mathscr {A}_{2n+1}\left( {w}\right) {\textbf{K}}\mathscr {B}_{2n+1}\left( {w}\right) . \end{aligned}$$

(15.49)

By virtue of (15.48), (15.34), (15.44), and (15.39), we discern

$$\begin{aligned} {\textbf{K}} =\sqrt{B}^{\mathord {+}}\left[ {\phi \left( {w}\right) -E\psi \left( {w}\right) } \right] \left[ {\phi \left( {w}\right) -E^*\psi \left( {w}\right) } \right] ^{\mathord {+}}\sqrt{B} =PCP. \end{aligned}$$

(15.50)

The application of Lemma 6.9 to the sequence $(s_j)_{j=0}^{\infty }\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,\infty }$ yields

$$\begin{aligned} \mathscr {A}_{2n}\left( {w}\right)&=\mathscr {A}_{2n+1}\left( {w}\right) ,&\mathscr {B}_{2n}\left( {w}\right)&=\mathscr {B}_{2n+1}\left( {w}\right) ,{} & {} \text {and}&\mathscr {C}_{2n}\left( {w}\right)&=\mathscr {C}_{2n+1}\left( {w}\right) . \end{aligned}$$

(15.51)

From Proposition 6.10 we get $\mathcal {N}\left( {\mathscr {A}_{2n}\left( {w}\right) }\right) =\mathcal {N}\left( {\mathfrak {h}_{2n}}\right)$ and $\mathcal {R}\left( {\mathscr {B}_{2n}\left( {w}\right) }\right) =\mathcal {R}\left( {\mathfrak {h}_{2n}}\right)$. According to (15.40), we have $P=\mathbb {P}_{\mathcal {R}\left( {\mathfrak {h}_{2n}}\right) }$ and $Q=\mathbb {P}_{\mathcal {N}\left( {\mathfrak {h}_{2n}}\right) }$. Clearly, then $P\mathscr {B}_{2n}\left( {w}\right) =\mathscr {B}_{2n}\left( {w}\right)$ follows. By virtue of Remark A.8, we see furthermore $\mathcal {R}\left( {P}\right) =\mathcal {R}\left( {\mathfrak {h}_{2n}}\right)$ and $\mathcal {R}\left( {Q}\right) =\mathcal {N}\left( {\mathfrak {h}_{2n}}\right)$. Hence, we get $\mathscr {A}_{2n}\left( {w}\right) Q=O_{{q\times q}}$. Remark A.7 provides $\mathcal {N}\left( {\mathfrak {h}_{2n}^*}\right) =\mathcal {R}\left( {\mathfrak {h}_{2n}}\right) ^\bot$. Since, in view of $(s_j)_{j=0}^{\infty }\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,\infty }$, Remark 5.3 yields $\mathfrak {h}_{2n}^*=\mathfrak {h}_{2n}$, then $\mathcal {N}\left( {\mathfrak {h}_{2n}}\right) =\mathcal {R}\left( {\mathfrak {h}_{2n}}\right) ^\bot$ follows. Thus, Remark A.9 yields $P+Q=I_{q}$, implying $\mathscr {A}_{2n}\left( {w}\right) P=\mathscr {A}_{2n}\left( {w}\right)$. Consequently, we infer $\mathscr {A}_{2n}\left( {w}\right) PCP\mathscr {B}_{2n}\left( {w}\right) =\mathscr {A}_{2n}\left( {w}\right) C\mathscr {B}_{2n}\left( {w}\right)$. Using additionally (15.33), (15.51), (15.50), (15.49), and (15.46), we conclude

$$\begin{aligned}\begin{aligned} X&=\mathscr {C}_{2n}\left( {w}\right) +\left( {w-\overline{w}}\right) ^{-1}\mathscr {A}_{2n}\left( {w}\right) C\mathscr {B}_{2n}\left( {w}\right) \\&=\mathscr {C}_{2n}\left( {w}\right) +\left( {w-\overline{w}}\right) ^{-1}\mathscr {A}_{2n}\left( {w}\right) PCP\mathscr {B}_{2n}\left( {w}\right) \\&=\mathscr {C}_{2n+1}\left( {w}\right) +\left( {w-\overline{w}}\right) ^{-1}\mathscr {A}_{2n+1}\left( {w}\right) {\textbf{K}}\mathscr {B}_{2n+1}\left( {w}\right) \\&=-\left[ {\mathfrak {a}_{n}\left( {w}\right) \mathfrak {h}_{2n}^{\mathord {+}}S+\mathfrak {a}_{n+1}\left( {w}\right) T} \right] {\mathfrak {R}}^{-1}=-\left[ {\textbf{a}_{n}\left( {w}\right) \mathfrak {h}_{2n}^{\mathord {+}}S+\textbf{a}_{n+1}\left( {w}\right) T} \right] {\textbf{R}}^{-1}. \end{aligned}\end{aligned}$$

Using additionally $\det {\textbf{R}} \ne 0$, (15.45), (14.19), and Lemma 14.2(a), we then obtain

$$\begin{aligned}&\begin{bmatrix}X\\ I_{q}\end{bmatrix} =\begin{bmatrix} -\left[ {\textbf{a}_{n}\left( {w}\right) \mathfrak {h}_{2n}^{\mathord {+}}S+\textbf{a}_{n+1}\left( {w}\right) T} \right] {\textbf{R}}^{-1}\\ {\textbf{R}} {\textbf{R}}^{-1}\end{bmatrix}\nonumber \\&\quad =\begin{bmatrix} -\textbf{a}_{n}\left( {w}\right) \mathfrak {h}_{2n}^{\mathord {+}}S-\textbf{a}_{n+1}\left( {w}\right) T\\ \textbf{b}_{n}\left( {w}\right) \mathfrak {h}_{2n}^{\mathord {+}}S+\textbf{b}_{n+1}\left( {w}\right) T\end{bmatrix} {\textbf{R}}^{-1}\nonumber \\&\quad =\begin{bmatrix} -\textbf{a}_{n}\left( {w}\right) \mathfrak {k}_{2n}^{\mathord {+}}&{}-\textbf{a}_{n+1}\left( {w}\right) \\ \textbf{b}_{n}\left( {w}\right) \mathfrak {k}_{2n}^{\mathord {+}}&{}\textbf{b}_{n+1}\left( {w}\right) \end{bmatrix}\begin{bmatrix} S\\ T\end{bmatrix}{\textbf{R}}^{-1}=\mathfrak {V}^{\left( {\alpha }\right) }_{2n+1}\left( {w}\right) \begin{bmatrix} S\\ T\end{bmatrix}{\textbf{R}}^{-1}. \end{aligned}$$

(15.52)

Part 3: Obviously, the matrix ${\mathbb {D}}_{\mathord {\circ }}:=\bigl [{\begin{matrix}\eta I_{q}&{}\mathfrak {k}_{2n+2}\\ O_{{q\times q}}&{}I_{q}\end{matrix}}\bigr ]$ is invertible with

(15.53)

From (15.24) and (15.17), we see ${{\,\textrm{rank}\,}}\bigl [{\begin{matrix}S_{\mathord {\circ }}\\ T_{\mathord {\circ }}\end{matrix}}\bigr ]=q$. Let

$$\begin{aligned} {\mathbb {D}}_{\mathord {\circ }}^{-1}\begin{bmatrix} S_{\mathord {\circ }}\\ T_{\mathord {\circ }}\end{bmatrix} {\textbf{R}}_{\mathord {\circ }}^{-1}= \begin{bmatrix} Y_{\mathord {\circ }}\\ Z_{\mathord {\circ }}\end{bmatrix} \end{aligned}$$

(15.54)

be the ${q\times q}$ block representation of ${\mathbb {D}}_{\mathord {\circ }}^{-1}\bigl [{\begin{matrix}S_{\mathord {\circ }}\\ T_{\mathord {\circ }}\end{matrix}}\bigr ]{\textbf{R}}_{\mathord {\circ }}^{-1}$. Then

$$\begin{aligned} {{\,\textrm{rank}\,}}\begin{bmatrix}Y_{\mathord {\circ }}\\ Z_{\mathord {\circ }}\end{bmatrix} ={{\,\textrm{rank}\,}}\begin{bmatrix}S_{\mathord {\circ }}\\ T_{\mathord {\circ }}\end{bmatrix} =q. \end{aligned}$$

(15.55)

Lemma 15.2 shows $\mathfrak {V}^{\left( {\alpha }\right) }_{2n+2}\left( {w}\right) {\mathbb {D}}_{\mathord {\circ }}=\bigl [{\begin{matrix}\eta I_{q}&{}s_{0}\\ O_{{q\times q}}&{} I_{q}\end{matrix}}\bigr ]\mathring{\mathbb {V}}_{2n+1}\left( {w}\right)$. Hence, using additionally (15.54) and (15.32), we deduce

$$\begin{aligned}\begin{aligned} \begin{bmatrix} \eta I_{q}&{}s_{0}\\ O_{{q\times q}}&{} I_{q}\end{bmatrix} \mathring{\mathbb {V}}_{2n+1}\left( {w}\right) \begin{bmatrix}Y_{\mathord {\circ }}\\ Z_{\mathord {\circ }}\end{bmatrix}&=\mathfrak {V}^{\left( {\alpha }\right) }_{2n+2}\left( {w}\right) \begin{bmatrix} S_{\mathord {\circ }}\\ T_{\mathord {\circ }}\end{bmatrix} {\textbf{R}}_{\mathord {\circ }}^{-1}\\&= \begin{bmatrix} \eta X+s_{0}\\ I_{q}\end{bmatrix} = \begin{bmatrix} \eta I_{q}&{}s_{0}\\ O_{{q\times q}}&{} I_{q}\end{bmatrix} \begin{bmatrix}X\\ I_{q}\end{bmatrix}. \end{aligned}\end{aligned}$$

Since the matrix $\bigl [{\begin{matrix}\eta I_{q}&{}s_{0}\\ O_{{q\times q}}&{} I_{q}\end{matrix}}\bigr ]$ is obviously invertible, then

$$\begin{aligned} \mathring{\mathbb {V}}_{2n+1}\left( {w}\right) \begin{bmatrix}Y_{\mathord {\circ }}\\ Z_{\mathord {\circ }}\end{bmatrix} =\begin{bmatrix}X\\ I_{q}\end{bmatrix} \end{aligned}$$

(15.56)

follows. Using (15.20), (14.19), and Remark A.10, we can infer

$$\begin{aligned} P_{\mathord {\circ }}=\mathbb {P}_{\mathcal {R}\left( {\mathfrak {k}_{2n+1}}\right) } =\mathfrak {k}_{2n+1}\mathfrak {k}_{2n+1}^{\mathord {+}}. \end{aligned}$$

(15.57)

Lemma 15.1 then shows

$$\begin{aligned} \mathfrak {V}^{\left( {\alpha }\right) }_{2n+1}\left( {w}\right) \begin{bmatrix} P_{\mathord {\circ }}&{}O_{{q\times q}}\\ \mathfrak {k}_{2n+1}^{\mathord {+}}&{}I_{q}\end{bmatrix} =\mathring{\mathbb {V}}_{2n+1}\left( {w}\right) . \end{aligned}$$

Hence, using additionally (15.52) and (15.56), we conclude

$$\begin{aligned} \mathfrak {V}^{\left( {\alpha }\right) }_{2n+1}\left( {w}\right) \begin{bmatrix}S\\ T\end{bmatrix}{\textbf{R}}^{-1}= & {} \begin{bmatrix}X\\ I_{q}\end{bmatrix} =\mathring{\mathbb {V}}_{2n+1}\left( {w}\right) \begin{bmatrix}Y_{\mathord {\circ }}\\ Z_{\mathord {\circ }}\end{bmatrix} \nonumber \\= & {} \mathfrak {V}^{\left( {\alpha }\right) }_{2n+1}\left( {w}\right) \begin{bmatrix} P_{\mathord {\circ }}&{}O_{{q\times q}}\\ \mathfrak {k}_{2n+1}^{\mathord {+}}&{}I_{q}\end{bmatrix} \begin{bmatrix}Y_{\mathord {\circ }}\\ Z_{\mathord {\circ }}\end{bmatrix}. \end{aligned}$$

(15.58)

Part 4: Regarding (15.54) and (15.53), we have

$$\begin{aligned} Y_{\mathord {\circ }}&=\eta ^{-1}\left( { S_{\mathord {\circ }}-\mathfrak {k}_{2n+2} T_{\mathord {\circ }}}\right) {\textbf{R}}_{\mathord {\circ }}^{-1}{} & {} \text {and}&Z_{\mathord {\circ }}&=T_{\mathord {\circ }}{\textbf{R}}_{\mathord {\circ }}^{-1}. \end{aligned}$$

(15.59)

In view of $(s_j)_{j=0}^{\infty }\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,\infty ,\alpha }$, Remark 9.3 yields $\mathfrak {k}_{2n+1}\mathfrak {k}_{2n+1}^{\mathord {+}}\mathfrak {k}_{2n+2}=\mathfrak {k}_{2n+2}$. By virtue of (15.57), therefore $P_{\mathord {\circ }}\mathfrak {k}_{2n+2}=\mathfrak {k}_{2n+2}$ follows. Taking additionally into account (15.27), from the first identity in (15.59) we then obtain

$$\begin{aligned} \begin{aligned} P_{\mathord {\circ }}Y_{\mathord {\circ }}=\eta ^{-1}\left( {P_{\mathord {\circ }}S_{\mathord {\circ }}-P_{\mathord {\circ }}\mathfrak {k}_{2n+2}T_{\mathord {\circ }}}\right) {\textbf{R}}_{\mathord {\circ }}^{-1}=\eta ^{-1}\left( {S_{\mathord {\circ }}-\mathfrak {k}_{2n+2}T_{\mathord {\circ }}}\right) {\textbf{R}}_{\mathord {\circ }}^{-1}=Y_{\mathord {\circ }}. \end{aligned}\end{aligned}$$

(15.60)

Let $\textbf{T}^{\left( {\alpha }\right) }_{2n+1}:\mathbb {C}\rightarrow \mathbb {C}^{{q\times q}}$ be defined by (15.1). Then, according to Lemma 15.3, we get $\mathfrak {W}^{\left( {\alpha }\right) }_{2n+1}\left( {w}\right) \mathfrak {V}^{\left( {\alpha }\right) }_{2n+1}\left( {w}\right) =\Delta$, where

$$\begin{aligned} \Delta :={{\,\textrm{diag}\,}}\left( {\mathfrak {k}_{2n}\mathfrak {k}_{2n}^{\mathord {+}},\eta ^{n+1}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) +\mathfrak {k}_{2n}^{\mathord {+}}\mathfrak {k}_{2n}+\textbf{T}^{\left( {\alpha }\right) }_{2n+1}\left( {w}\right) }\right) . \end{aligned}$$

Consequently, in view of (15.58), we infer

$$\begin{aligned} \Delta \begin{bmatrix}S\\ T\end{bmatrix}{\textbf{R}}^{-1}=\Delta \begin{bmatrix}P_{\mathord {\circ }}&{}O_{{q\times q}}\\ \mathfrak {k}_{2n+1}^{\mathord {+}}&{}I_{q}\end{bmatrix}\begin{bmatrix}Y_{\mathord {\circ }}\\ Z_{\mathord {\circ }}\end{bmatrix}. \end{aligned}$$

Therefore,

$$\begin{aligned} \mathfrak {k}_{2n}\mathfrak {k}_{2n}^{\mathord {+}}S{\textbf{R}}^{-1}=\mathfrak {k}_{2n}\mathfrak {k}_{2n}^{\mathord {+}}P_{\mathord {\circ }}Y_{\mathord {\circ }}\end{aligned}$$

(15.61)

and

$$\begin{aligned}{} & {} \left[ {\eta ^{n+1}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) +\mathfrak {k}_{2n}^{\mathord {+}}\mathfrak {k}_{2n}+\textbf{T}^{\left( {\alpha }\right) }_{2n+1}\left( {w}\right) } \right] T{\textbf{R}}^{-1}\nonumber \\{} & {} \quad =\left[ {\eta ^{n+1}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) +\mathfrak {k}_{2n}^{\mathord {+}}\mathfrak {k}_{2n}+\textbf{T}^{\left( {\alpha }\right) }_{2n+1}\left( {w}\right) } \right] \left( {\mathfrak {k}_{2n+1}^{\mathord {+}}Y_{\mathord {\circ }}+Z_{\mathord {\circ }}}\right) . \end{aligned}$$

(15.62)

In view of $(s_j)_{j=0}^{\infty }\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,\infty ,\alpha }$, Remark 9.3 yields (14.70). If $n\ge 1$, then from (15.1) and (14.70) we can see that

$$\begin{aligned} \textbf{T}^{\left( {\alpha }\right) }_{2n+1}\left( {w}\right) \mathfrak {k}_{2n+1}^{\mathord {+}}=\sum _{k=0}^{n-1}\eta ^{n-k}\left( {\mathfrak {k}_{2k}^{\mathord {+}}\mathfrak {k}_{2k}\mathfrak {k}_{2n+1}^{\mathord {+}}-\mathfrak {k}_{2k+1}^{\mathord {+}}\mathfrak {k}_{2k+1}\mathfrak {k}_{2n+1}^{\mathord {+}}}\right) =O_{{q\times q}}\end{aligned}$$

and, for each $m\in \left\{ {2n,2n+1} \right\}$, moreover,

$$\begin{aligned} \mathfrak {k}_{m}\textbf{T}^{\left( {\alpha }\right) }_{2n+1}\left( {w}\right) =\sum _{k=0}^{n-1}\eta ^{n-k}\left( {\mathfrak {k}_{m}\mathfrak {k}_{2k}^{\mathord {+}}\mathfrak {k}_{2k}-\mathfrak {k}_{m}\mathfrak {k}_{2k+1}^{\mathord {+}}\mathfrak {k}_{2k+1}}\right) =O_{{q\times q}}\end{aligned}$$

are fulfilled. Regarding (15.1) also in the case $n=0$, we can conclude then in general

$$\begin{aligned} \textbf{T}^{\left( {\alpha }\right) }_{2n+1}\left( {w}\right) \mathfrak {k}_{2n+1}^{\mathord {+}}&=O_{{q\times q}},&\mathfrak {k}_{2n}\textbf{T}^{\left( {\alpha }\right) }_{2n+1}\left( {w}\right)&=O_{{q\times q}},&\mathfrak {k}_{2n+1}\textbf{T}^{\left( {\alpha }\right) }_{2n+1}\left( {w}\right)&=O_{{q\times q}}. \end{aligned}$$

(15.63)

Using (15.40), (14.19), and Remark A.10, we can infer $P=\mathfrak {k}_{2n}\mathfrak {k}_{2n}^{\mathord {+}}$. Regarding additionally (14.20), the second identity in (15.63), (14.70), and (15.47), then

$$\begin{aligned}&\left( {\left[ {\eta ^{n+1}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) +\mathfrak {k}_{2n}^{\mathord {+}}\mathfrak {k}_{2n}+\textbf{T}^{\left( {\alpha }\right) }_{2n+1}\left( {w}\right) } \right] T{\textbf{R}}^{-1}}\right) ^*\left( {\mathfrak {k}_{2n}\mathfrak {k}_{2n}^{\mathord {+}}S{\textbf{R}}^{-1}}\right) \nonumber \\&={\textbf{R}}^{-*}T^*\left[ {\eta ^{n+1}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) +\mathfrak {k}_{2n}^{\mathord {+}}\mathfrak {k}_{2n}+\textbf{T}^{\left( {\alpha }\right) }_{2n+1}\left( {w}\right) } \right] ^*\mathfrak {k}_{2n}^*\mathfrak {k}_{2n}^{\mathord {+}}S{\textbf{R}}^{-1}\nonumber \\&={\textbf{R}}^{-*}T^*\left[ {\eta ^{n+1}\mathfrak {k}_{2n}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) +\mathfrak {k}_{2n}\mathfrak {k}_{2n}^{\mathord {+}}\mathfrak {k}_{2n}+\mathfrak {k}_{2n}\textbf{T}^{\left( {\alpha }\right) }_{2n+1}\left( {w}\right) } \right] ^*\mathfrak {k}_{2n}^{\mathord {+}}S{\textbf{R}}^{-1}\nonumber \\&={\textbf{R}}^{-*}T^*\left[ {\eta ^{n+1}\left( {\mathfrak {k}_{2n}-\mathfrak {k}_{2n}\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) +\mathfrak {k}_{2n}} \right] ^*\mathfrak {k}_{2n}^{\mathord {+}}S{\textbf{R}}^{-1}={\textbf{R}}^{-*}T^*\mathfrak {k}_{2n}^*\mathfrak {k}_{2n}^{\mathord {+}}S{\textbf{R}}^{-1}\nonumber \\&={\textbf{R}}^{-*}T^*\mathfrak {k}_{2n}\mathfrak {k}_{2n}^{\mathord {+}}S{\textbf{R}}^{-1}={\textbf{R}}^{-*}T^*PS{\textbf{R}}^{-1}={\textbf{R}}^{-*}T^*S{\textbf{R}}^{-1}\end{aligned}$$

(15.64)

follows. In view of (15.57), we can infer from (14.70) that

$$\begin{aligned} \mathfrak {k}_{\ell }\mathfrak {k}_{\ell }^{\mathord {+}}P_{\mathord {\circ }}=P_{\mathord {\circ }}\quad \text {for all }\ell \in \mathbb {Z}_{0,2n+1}. \end{aligned}$$

(15.65)

From (15.61) we thus can conclude

$$\begin{aligned} \mathfrak {k}_{2n}\mathfrak {k}_{2n}^{\mathord {+}}S{\textbf{R}}^{-1}=\mathfrak {k}_{2n}\mathfrak {k}_{2n}^{\mathord {+}}P_{\mathord {\circ }}Y_{\mathord {\circ }}=P_{\mathord {\circ }}Y_{\mathord {\circ }}. \end{aligned}$$

(15.66)

In view of (15.47) and $P=\mathfrak {k}_{2n}\mathfrak {k}_{2n}^{\mathord {+}}$, then $S{\textbf{R}}^{-1}=PS{\textbf{R}}^{-1}=\mathfrak {k}_{2n}\mathfrak {k}_{2n}^{\mathord {+}}S{\textbf{R}}^{-1}=P_{\mathord {\circ }}Y_{\mathord {\circ }}$ follows. Furthermore, (15.62), (14.70), and the first identity in (15.63) yield

$$\begin{aligned}&\left[ {\eta ^{n+1}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) +\mathfrak {k}_{2n}^{\mathord {+}}\mathfrak {k}_{2n}+\textbf{T}^{\left( {\alpha }\right) }_{2n+1}\left( {w}\right) } \right] T{\textbf{R}}^{-1}\nonumber \\&=\left[ {\eta ^{n+1}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) +\mathfrak {k}_{2n}^{\mathord {+}}\mathfrak {k}_{2n}+\textbf{T}^{\left( {\alpha }\right) }_{2n+1}\left( {w}\right) } \right] \left( {\mathfrak {k}_{2n+1}^{\mathord {+}}Y_{\mathord {\circ }}+Z_{\mathord {\circ }}}\right) \nonumber \\&=\eta ^{n+1}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) \mathfrak {k}_{2n+1}^{\mathord {+}}Y_{\mathord {\circ }}+\mathfrak {k}_{2n}^{\mathord {+}}\mathfrak {k}_{2n}\mathfrak {k}_{2n+1}^{\mathord {+}}Y_{\mathord {\circ }}+\textbf{T}^{\left( {\alpha }\right) }_{2n+1}\left( {w}\right) \mathfrak {k}_{2n+1}^{\mathord {+}}Y_{\mathord {\circ }}\nonumber \\&\qquad +\left[ {\eta ^{n+1}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) +\mathfrak {k}_{2n}^{\mathord {+}}\mathfrak {k}_{2n}+\textbf{T}^{\left( {\alpha }\right) }_{2n+1}\left( {w}\right) } \right] Z_{\mathord {\circ }}\nonumber \\&=\eta ^{n+1}\left( {\mathfrak {k}_{2n+1}^{\mathord {+}}-\mathfrak {k}_{2n+1}^{\mathord {+}}}\right) Y_{\mathord {\circ }}+\mathfrak {k}_{2n+1}^{\mathord {+}}Y_{\mathord {\circ }}\nonumber \\&\qquad +\left[ {\eta ^{n+1}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) +\mathfrak {k}_{2n}^{\mathord {+}}\mathfrak {k}_{2n}+\textbf{T}^{\left( {\alpha }\right) }_{2n+1}\left( {w}\right) } \right] Z_{\mathord {\circ }}\nonumber \\&=\mathfrak {k}_{2n+1}^{\mathord {+}}Y_{\mathord {\circ }}+\left[ {\eta ^{n+1}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) +\mathfrak {k}_{2n}^{\mathord {+}}\mathfrak {k}_{2n}+\textbf{T}^{\left( {\alpha }\right) }_{2n+1}\left( {w}\right) } \right] Z_{\mathord {\circ }}. \end{aligned}$$

(15.67)

In view of (14.20), we can infer from Remark A.12 easily $\left( {\mathfrak {k}_{2n+1}^{\mathord {+}}}\right) ^*=\mathfrak {k}_{2n+1}^{\mathord {+}}$ and $\left( {\mathfrak {k}_{j}^{\mathord {+}}\mathfrak {k}_{j}}\right) ^*=\mathfrak {k}_{j}\mathfrak {k}_{j}^{\mathord {+}}$ for all $j\in \mathbb {N}_0$. Regarding (15.65), in particular $\left( {\mathfrak {k}_{2n}^{\mathord {+}}\mathfrak {k}_{2n}}\right) ^*P_{\mathord {\circ }}=P_{\mathord {\circ }}$ and $\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) ^*P_{\mathord {\circ }}=O_{{q\times q}}$ follow. Using (15.57), (14.20), and the last identity in (15.63), we obtain $\left[ {\textbf{T}^{\left( {\alpha }\right) }_{2n+1}\left( {w}\right) } \right] ^*P_{\mathord {\circ }}=\left[ {\mathfrak {k}_{2n+1}\textbf{T}^{\left( {\alpha }\right) }_{2n+1}\left( {w}\right) } \right] ^*\mathfrak {k}_{2n+1}^{\mathord {+}}=O_{{q\times q}}$. Taking additionally into account (15.67), (15.66), and (15.60), we then get

$$\begin{aligned} \begin{aligned}&\left( {\left[ {\eta ^{n+1}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) +\mathfrak {k}_{2n}^{\mathord {+}}\mathfrak {k}_{2n}+\textbf{T}^{\left( {\alpha }\right) }_{2n+1}\left( {w}\right) } \right] T{\textbf{R}}^{-1}}\right) ^*\left( {\mathfrak {k}_{2n}\mathfrak {k}_{2n}^{\mathord {+}}S{\textbf{R}}^{-1}}\right) \\&=\left( {\mathfrak {k}_{2n+1}^{\mathord {+}}Y_{\mathord {\circ }}+\left[ {\eta ^{n+1}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) +\mathfrak {k}_{2n}^{\mathord {+}}\mathfrak {k}_{2n}+\textbf{T}^{\left( {\alpha }\right) }_{2n+1}\left( {w}\right) } \right] Z_{\mathord {\circ }}}\right) ^*\left( {P_{\mathord {\circ }}Y_{\mathord {\circ }}}\right) \\&=Y_{\mathord {\circ }}^*\left( {\mathfrak {k}_{2n+1}^{\mathord {+}}}\right) ^*P_{\mathord {\circ }}Y_{\mathord {\circ }}+Z_{\mathord {\circ }}^*\left[ {\eta ^{n+1}\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) +\mathfrak {k}_{2n}^{\mathord {+}}\mathfrak {k}_{2n}+\textbf{T}^{\left( {\alpha }\right) }_{2n+1}\left( {w}\right) } \right] ^*P_{\mathord {\circ }}Y_{\mathord {\circ }}\\&=Y_{\mathord {\circ }}^*\left( {\mathfrak {k}_{2n+1}^{\mathord {+}}}\right) ^*P_{\mathord {\circ }}Y_{\mathord {\circ }}\\&\qquad +\overline{\eta }^{n+1}Z_{\mathord {\circ }}^*\left( {I_{q}-\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{0}}\right) ^*P_{\mathord {\circ }}Y_{\mathord {\circ }}+Z_{\mathord {\circ }}^*\left( {\mathfrak {k}_{2n}^{\mathord {+}}\mathfrak {k}_{2n}}\right) ^*P_{\mathord {\circ }}Y_{\mathord {\circ }}+Z_{\mathord {\circ }}^*\left[ {\textbf{T}^{\left( {\alpha }\right) }_{2n+1}\left( {w}\right) } \right] ^*P_{\mathord {\circ }}Y_{\mathord {\circ }}\\&=Y_{\mathord {\circ }}^*\mathfrak {k}_{2n+1}^{\mathord {+}}P_{\mathord {\circ }}Y_{\mathord {\circ }}+Z_{\mathord {\circ }}^*P_{\mathord {\circ }}Y_{\mathord {\circ }}=Y_{\mathord {\circ }}^*\mathfrak {k}_{2n+1}^{\mathord {+}}Y_{\mathord {\circ }}+Z_{\mathord {\circ }}^*Y_{\mathord {\circ }}. \end{aligned}\end{aligned}$$

(15.68)

Combining (15.64) and (15.68), we obtain ${\textbf{R}}^{-*}T^*S{\textbf{R}}^{-1}=Y_{\mathord {\circ }}^*\mathfrak {k}_{2n+1}^{\mathord {+}}Y_{\mathord {\circ }}+Z_{\mathord {\circ }}^*Y_{\mathord {\circ }}$ and, thus, we conclude $Z_{\mathord {\circ }}^*Y_{\mathord {\circ }}={\textbf{R}}^{-*}T^*S{\textbf{R}}^{-1}-Y_{\mathord {\circ }}^*\mathfrak {k}_{2n+1}^{\mathord {+}}Y_{\mathord {\circ }}$. Since $\left( {\mathfrak {k}_{2n+1}^{\mathord {+}}}\right) ^*=\mathfrak {k}_{2n+1}^{\mathord {+}}$ implies $\Im \left( {Y_{\mathord {\circ }}^*\mathfrak {k}_{2n+1}^{\mathord {+}}Y_{\mathord {\circ }}}\right) =O_{{q\times q}}$ and since (15.44) and (15.37) show $\Im \left( {T^*S}\right) \in \mathbb {C}_\succcurlyeq ^{{q\times q}}$, the application of Remarks A.1, A.2, and A.6 then yields

$$\begin{aligned} \Im \left( {Z_{\mathord {\circ }}^*Y_{\mathord {\circ }}}\right) =\Im \left( {{\textbf{R}}^{-*}T^*S{\textbf{R}}^{-1}}\right) -\Im \left( {Y_{\mathord {\circ }}^*\mathfrak {k}_{2n+1}^{\mathord {+}}Y_{\mathord {\circ }}}\right) ={\textbf{R}}^{-*}\Im \left( {T^*S}\right) {\textbf{R}}^{-1}\in \mathbb {C}_\succcurlyeq ^{{q\times q}}. \end{aligned}$$

(15.69)

Because of (14.20), we have $\Im \left( {T_{\mathord {\circ }}^*\mathfrak {k}_{2n+2}T_{\mathord {\circ }}}\right) =O_{{q\times q}}$. From (15.24) and (15.17) we see $\Im \left( {T_{\mathord {\circ }}^*S_{\mathord {\circ }}}\right) \in \mathbb {C}_\succcurlyeq ^{{q\times q}}$. Hence, using (15.59) as well as Remarks A.2, A.1, and A.6, we infer

$$\begin{aligned} \begin{aligned} \Im \left( {\eta Z_{\mathord {\circ }}^*Y_{\mathord {\circ }}}\right)&=\Im \left( {\eta \left( { T_{\mathord {\circ }}{\textbf{R}}_{\mathord {\circ }}^{-1}}\right) ^*\left[ {\eta ^{-1}\left( { S_{\mathord {\circ }}-\mathfrak {k}_{2n+2} T_{\mathord {\circ }}}\right) {\textbf{R}}_{\mathord {\circ }}^{-1}} \right] }\right) \\&=\Im \left( {{\textbf{R}}_{\mathord {\circ }}^{-*}T_{\mathord {\circ }}^*\left( { S_{\mathord {\circ }}-\mathfrak {k}_{2n+2} T_{\mathord {\circ }}}\right) {\textbf{R}}_{\mathord {\circ }}^{-1}}\right) \\&={\textbf{R}}_{\mathord {\circ }}^{-*}\left[ {\Im \left( { T_{\mathord {\circ }}^*S_{\mathord {\circ }}}\right) -\Im \left( { T_{\mathord {\circ }}^*\mathfrak {k}_{2n+2} T_{\mathord {\circ }}}\right) } \right] {\textbf{R}}_{\mathord {\circ }}^{-1}\\&={\textbf{R}}_{\mathord {\circ }}^{-*}\Im \left( { T_{\mathord {\circ }}^*S_{\mathord {\circ }}}\right) {\textbf{R}}_{\mathord {\circ }}^{-1}\in \mathbb {C}_\succcurlyeq ^{{q\times q}}. \end{aligned}\end{aligned}$$

(15.70)

Part 5: Let $\pi _{\mathord {\circ }},\rho _{\mathord {\circ }}:\mathbb {C}\rightarrow \mathbb {C}^{{q\times q}}$ be defined by

$$\begin{aligned} \pi _{\mathord {\circ }}\left( {z}\right) :=Y_{\mathord {\circ }}\text { and } \rho _{\mathord {\circ }}\left( {z}\right) :=\frac{w-z}{\Im w}H_{\mathord {\circ }}Y_{\mathord {\circ }}+Z_{\mathord {\circ }}, \text { where } H_{\mathord {\circ }}:=\left( {Y_{\mathord {\circ }}^{\mathord {+}}}\right) ^*\Im \left( {Z_{\mathord {\circ }}^*Y_{\mathord {\circ }}}\right) Y_{\mathord {\circ }}^{\mathord {+}}. \nonumber \\ \end{aligned}$$

(15.71)

In view of (15.69) and (15.71), we can apply Lemma 8.5 and it follows that $\pi _{\mathord {\circ }}$ and $\rho _{\mathord {\circ }}$ are both holomorphic in $\mathbb {C}$ fulfilling

$$\begin{aligned} \begin{bmatrix}\pi _{\mathord {\circ }}\left( {w}\right) \\ \rho _{\mathord {\circ }}\left( {w}\right) \end{bmatrix} =\begin{bmatrix}Y_{\mathord {\circ }}\\ Z_{\mathord {\circ }}\end{bmatrix} \end{aligned}$$

(15.72)

as well as

$$\begin{aligned} {{\,\textrm{rank}\,}}\begin{bmatrix}\pi _{\mathord {\circ }}\left( {z}\right) \\ \rho _{\mathord {\circ }}\left( {z}\right) \end{bmatrix}&={{\,\textrm{rank}\,}}\begin{bmatrix}Y_{\mathord {\circ }}\\ Z_{\mathord {\circ }}\end{bmatrix}, \end{aligned}$$

(15.73)

$$\begin{aligned} \Im \left( {\left[ {\rho _{\mathord {\circ }}\left( {z}\right) } \right] ^*\pi _{\mathord {\circ }}\left( {z}\right) }\right)&=\frac{\Im z}{\Im w}\Im \left( {Z_{\mathord {\circ }}^*Y_{\mathord {\circ }}}\right) , \end{aligned}$$

(15.74)

and

$$\begin{aligned} \Im \left( {\left( {z-\alpha }\right) \left[ {\rho _{\mathord {\circ }}\left( {z}\right) } \right] ^*\pi _{\mathord {\circ }}\left( {z}\right) }\right)&=\frac{\Im z}{\Im w}\Im \left( {\eta Z_{\mathord {\circ }}^*Y_{\mathord {\circ }}}\right) \end{aligned}$$

(15.75)

for all $z\in \mathbb {C}$. Let $\phi _{\mathord {\bullet }},\psi _{\mathord {\bullet }}:\mathbb {C}\backslash {[\alpha ,\infty )}\rightarrow \mathbb {C}^{{q\times q}}$ be defined by

$$\begin{aligned} \phi _{\mathord {\bullet }}\left( {z}\right) :=\pi _{\mathord {\circ }}\left( {z}\right) \quad \text {and}\quad \psi _{\mathord {\bullet }}\left( {z}\right) :=\rho _{\mathord {\circ }}\left( {z}\right) . \end{aligned}$$

(15.76)

Clearly, $\mathcal {D}_{\mathord {\bullet }}:=\emptyset$ is a discrete subset of $\mathbb {C}\backslash {[\alpha ,\infty )}$. Keeping in mind that $\pi _{\mathord {\circ }}$ and $\rho _{\mathord {\circ }}$ are holomorphic in $\mathbb {C}$, we see that $\phi _{\mathord {\bullet }}$ and $\psi _{\mathord {\bullet }}$ are holomorphic and, in particular, meromorphic in $\mathbb {C}\backslash {[\alpha ,\infty )}$. For all $z\in \mathbb {C}\backslash {[\alpha ,\infty )}$, equations (15.76), (15.73), and (15.55) imply ${{\,\textrm{rank}\,}}\bigl [{\begin{matrix}\phi _{\mathord {\bullet }}\left( {z}\right) \\ \psi _{\mathord {\bullet }}\left( {z}\right) \end{matrix}}\bigr ]={{\,\textrm{rank}\,}}\bigl [{\begin{matrix}\pi _{\mathord {\circ }}\left( {z}\right) \\ \rho _{\mathord {\circ }}\left( {z}\right) \end{matrix}}\bigr ]={{\,\textrm{rank}\,}}\bigl [{\begin{matrix}Y_{\mathord {\circ }}\\ Z_{\mathord {\circ }}\end{matrix}}\bigr ]=q$. Regarding $\Im w\in (0,\infty )$, for all $z\in \mathbb {C}\backslash \mathbb {R}$, moreover (15.76), (15.74), and (15.69) yield

$$\begin{aligned} \frac{1}{\Im z}\Im \left( {\left[ {\psi _{\mathord {\bullet }}\left( {z}\right) } \right] ^*\phi _{\mathord {\bullet }}\left( {z}\right) }\right) =\frac{1}{\Im z}\Im \left( {\left[ {\rho _{\mathord {\circ }}\left( {z}\right) } \right] ^*\pi _{\mathord {\circ }}\left( {z}\right) }\right) =\frac{1}{\Im w}\Im \left( {Z_{\mathord {\circ }}^*Y_{\mathord {\circ }}}\right) \in \mathbb {C}_\succcurlyeq ^{{q\times q}}, \end{aligned}$$

whereas (15.76), (15.75), and (15.70) provide

$$\begin{aligned}\begin{aligned} \left( {\Im z}\right) ^{-1}\Im \left( {\left( {z-\alpha }\right) \left[ {\psi _{\mathord {\bullet }}\left( {z}\right) } \right] ^*\phi _{\mathord {\bullet }}\left( {z}\right) }\right)&=\left( {\Im z}\right) ^{-1}\Im \left( {\left( {z-\alpha }\right) \left[ {\rho _{\mathord {\circ }}\left( {z}\right) } \right] ^*\pi _{\mathord {\circ }}\left( {z}\right) }\right) \\&=\left( {\Im w}\right) ^{-1}\Im \left( {\eta Z_{\mathord {\circ }}^*Y_{\mathord {\circ }}}\right) \in \mathbb {C}_\succcurlyeq ^{{q\times q}}. \end{aligned} \end{aligned}$$

In view of Remark 8.2, then, according to Definition 8.1, the pair $\left( {\phi _{\mathord {\bullet }}};{\psi _{\mathord {\bullet }}}\right)$ belongs to $\mathcal{P}\mathcal{S}_{q}\left( {\mathbb {C}\backslash {[\alpha ,\infty )}}\right)$ and the set $\mathcal {D}_{\mathord {\bullet }}$ belongs to $\mathscr {D}_{\mathord {\bullet }}\left( {\phi _{\mathord {\bullet }},\psi _{\mathord {\bullet }}}\right)$. Regarding Notation 7.1, Definition 5.2, and (5.2), we have $\mathfrak {h}_{{\alpha ,2n}}=L_{{\alpha ,n}}$. Using additionally (15.76), (15.20), (15.71), and (15.60), we can conclude $\mathbb {P}_{\mathcal {R}\left( {L_{{\alpha ,n}}}\right) }\phi _{\mathord {\bullet }}\left( {z}\right) =\mathbb {P}_{\mathcal {R}\left( {\mathfrak {h}_{{\alpha ,2n}}}\right) }\pi _{\mathord {\circ }}\left( {z}\right) =P_{\mathord {\circ }}Y_{\mathord {\circ }}=Y_{\mathord {\circ }}=\pi _{\mathord {\circ }}\left( {z}\right) =\phi _{\mathord {\bullet }}\left( {z}\right)$ for all $z\in \mathbb {C}\backslash {[\alpha ,\infty )}$. By virtue of Notation 8.4, consequently $\left( {\phi _{\mathord {\bullet }}};{\psi _{\mathord {\bullet }}}\right) \in \mathcal {P}_{{\mathord {\bullet }},\alpha }\left[ {L_{{\alpha ,n}}} \right]$.

Part 6: From $\left( {\phi _{\mathord {\bullet }}};{\psi _{\mathord {\bullet }}}\right) \in \mathcal {P}_{{\mathord {\bullet }},\alpha }\left[ {L_{{\alpha ,n}}} \right]$, Theorem 10.18(a) and the notations therein, we see that $\det \left( {\tilde{\textbf{p}}_{2n+1}L_{{\alpha ,n}}^{\mathord {+}}\phi _{\mathord {\bullet }}+\tilde{\textbf{p}}_{2n+2}\psi _{\mathord {\bullet }}}\right)$ does not vanish identically and that

$$\begin{aligned} F :=-\left( {\tilde{\textbf{q}}_{2n+1}L_{{\alpha ,n}}^{\mathord {+}}\phi _{\mathord {\bullet }}+\tilde{\textbf{q}}_{2n+2}\psi _{\mathord {\bullet }}}\right) \left( {\tilde{\textbf{p}}_{2n+1}L_{{\alpha ,n}}^{\mathord {+}}\phi _{\mathord {\bullet }}+\tilde{\textbf{p}}_{2n+2}\psi _{\mathord {\bullet }}}\right) ^{-1}\end{aligned}$$

(15.77)

belongs to ${\mathcal {S}_{0,q}\left[ {\mathbb {C}\backslash {[\alpha ,\infty )};(s_j)_{j=0}^{2n+1},\preccurlyeq } \right] }$. Furthermore, regarding (15.56), (15.72), (15.76), and (14.4), we have

$$\begin{aligned}\begin{aligned} \begin{bmatrix}X\\ I_{q}\end{bmatrix}&=\mathring{\mathbb {V}}_{2n+1}\left( {w}\right) \begin{bmatrix}Y_{\mathord {\circ }}\\ Z_{\mathord {\circ }}\end{bmatrix} =\mathring{\mathbb {V}}_{2n+1}\left( {w}\right) \begin{bmatrix}\pi _{\mathord {\circ }}\left( {w}\right) \\ \rho _{\mathord {\circ }}\left( {w}\right) \end{bmatrix}\\&=\mathring{\mathbb {V}}_{2n+1}\left( {w}\right) \begin{bmatrix}\phi _{\mathord {\bullet }}\left( {w}\right) \\ \psi _{\mathord {\bullet }}\left( {w}\right) \end{bmatrix} = \begin{bmatrix} -\textbf{q}_{2n+1}\left( {w}\right) \mathfrak {k}_{2n+1}^{\mathord {+}}&{}-\textbf{q}_{2n+2}\left( {w}\right) \\ \textbf{p}_{2n+1}\left( {w}\right) \mathfrak {k}_{2n+1}^{\mathord {+}}&{}\textbf{p}_{2n+2}\left( {w}\right) \end{bmatrix}\begin{bmatrix}\phi _{\mathord {\bullet }}\left( {w}\right) \\ \psi _{\mathord {\bullet }}\left( {w}\right) \end{bmatrix}. \end{aligned}\end{aligned}$$

Since $\mathfrak {k}_{2n+1}=\mathfrak {h}_{{\alpha ,2n}}=L_{{\alpha ,n}}$ holds true because of (14.19), we consequently get $X=-\left[ {\tilde{\textbf{q}}_{2n+1}\left( {w}\right) L_{{\alpha ,n}}^{\mathord {+}}\phi _{\mathord {\bullet }}\left( {w}\right) +\tilde{\textbf{q}}_{2n+2}\left( {w}\right) \psi _{\mathord {\bullet }}\left( {w}\right) } \right]$ and $I_{q}=\tilde{\textbf{p}}_{2n+1}\left( {w}\right) L_{{\alpha ,n}}^{\mathord {+}}\phi _{\mathord {\bullet }}\left( {w}\right) +\tilde{\textbf{p}}_{2n+2}\left( {w}\right) \psi _{\mathord {\bullet }}\left( {w}\right)$. In particular, $\det \left( {\tilde{\textbf{p}}_{2n+1}\left( {w}\right) L_{{\alpha ,n}}^{\mathord {+}}\phi _{\mathord {\bullet }}\left( {w}\right) +\tilde{\textbf{p}}_{2n+2}\left( {w}\right) \psi _{\mathord {\bullet }}\left( {w}\right) }\right) \ne 0$. Taking additionally into account (15.77), we finally see $X=-\left[ {\tilde{\textbf{q}}_{2n+1}\left( {w}\right) L_{{\alpha ,n}}^{\mathord {+}}\phi _{\mathord {\bullet }}\left( {w}\right) +\tilde{\textbf{q}}_{2n+2}\left( {w}\right) \psi _{\mathord {\bullet }}\left( {w}\right) } \right] \left[ {\tilde{\textbf{p}}_{2n+1}\left( {w}\right) L_{{\alpha ,n}}^{\mathord {+}}\phi _{\mathord {\bullet }}\left( {w}\right) +\tilde{\textbf{p}}_{2n+2}\left( {w}\right) \psi _{\mathord {\bullet }}\left( {w}\right) } \right] ^{-1}=F\left( {w}\right)$. $\square$

Now we get our second main result, which describes the set of possible values of the functions corresponding to solutions of the Stieltjes moment problem in the case of an even number of prescribed matrix moments.

Theorem 15.5

Let $n\in \mathbb {N}_0$, let $(s_j)_{j=0}^{2n+1}\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,2n+1,\alpha }$, and let $w\in \Pi _{\mathord {+}}$. Then

$$\begin{aligned} \left\{ {F\left( {w}\right) }:{F\in {\mathcal {S}_{0,q}\left[ {\mathbb {C}\backslash {[\alpha ,\infty )};(s_j)_{j=0}^{2n+1},\preccurlyeq } \right] }}\right\} =\mathscr {K}_{2n}\left( {w}\right) \cap \mathscr {K}_{{{\mathord {\circ }},2n}}\left( {w}\right) . \end{aligned}$$

Proof

Combine Propositions 13.3 and 15.4. $\square$

16 The Case of a Single Prescribed Matrix Moment

At the end of this paper, we turn our attention to the case that only the matrix moment $s_{0}$ is prescribed. Note that this problem was already studied in [19]. Our approach to this problem is linked to the previous considerations. Let

$$\begin{aligned} \mathscr {H}_q:=\left\{ {X\in \mathbb {C}^{{q\times q}}}:{\Im X\in \mathbb {C}_\succcurlyeq ^{{q\times q}}}\right\} . \end{aligned}$$

(16.1)

For all $z\in \mathbb {C}$, let

$$\begin{aligned} \mathscr {S}_{q,\alpha }\left( {z}\right) :=\left\{ {X\in \mathbb {C}^{{q\times q}}}:{\Im \left( {\left( {z-\alpha }\right) X}\right) \in \mathbb {C}_\succcurlyeq ^{{q\times q}}}\right\} . \end{aligned}$$

(16.2)

Corollary 16.1

Let $F\in \mathcal {S}_{q}\left( {\mathbb {C}\backslash {[\alpha ,\infty )}}\right)$ and let $w\in \Pi _{\mathord {+}}$. Then $F\left( {w}\right) \in \mathscr {H}_q\cap \mathscr {S}_{q,\alpha }\left( {w}\right)$.

Proof

Regarding (16.1), (16.2), and the definition of the class $\mathcal {R}_{q}\left( {\Pi _{\mathord {+}}}\right)$, the assertion follows from [11, Prop. 4.3]. $\square$

Proposition 16.2

(cf. [19, Lem. 14.10]) Let $s_{0}\in \mathbb {C}_\succcurlyeq ^{{q\times q}}$, let $F\in {\mathcal {S}_{0,q}\left[ {\mathbb {C}\backslash {[\alpha ,\infty )};(s_j)_{j=0}^{0},\preccurlyeq } \right] }$, and let $w\in \Pi _{\mathord {+}}$. Then $(s_j)_{j=0}^{0}\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,0}$ and $F\left( {w}\right) \in \mathscr {K}_{0}\left( {w}\right) \cap \mathscr {S}_{q,\alpha }\left( {w}\right)$.

Proof

Since $s_{0}\in \mathbb {C}_\succcurlyeq ^{{q\times q}}$, we have $(s_j)_{j=0}^{0}\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,0,\alpha }$. Thus, we can apply Lemma 13.1 to obtain $(s_j)_{j=0}^{0}\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,0}$ and $F\left( {w}\right) \in \mathscr {K}_{0}\left( {w}\right)$. Furthermore, $F\in \mathcal {S}_{0,q}\left( {\mathbb {C}\backslash {[\alpha ,\infty )}}\right)$ and, in particular, $F\in \mathcal {S}_{q}\left( {\mathbb {C}\backslash {[\alpha ,\infty )}}\right)$. Consequently, we can apply Corollary 16.1 to obtain $F\left( {w}\right) \in \mathscr {S}_{q,\alpha }\left( {w}\right)$. $\square$

Lemma 16.3

Let $(s_j)_{j=0}^{\kappa }\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,\kappa ,\alpha }$ and let $z\in \mathbb {C}$. Then

$$\begin{aligned} \begin{bmatrix} \left( {z-\alpha }\right) I_{q}&{}s_{0}\\ O_{{q\times q}}&{}I_{q}\end{bmatrix}\mathring{\mathbb {V}}_{0}\left( {z}\right) =\left( {z-\alpha }\right) \begin{bmatrix} \mathfrak {k}_{0}\mathfrak {k}_{0}^{\mathord {+}}&{}O_{{q\times q}}\\ \mathfrak {k}_{0}^{\mathord {+}}&{}I_{q}\end{bmatrix}. \end{aligned}$$

(16.3)

Proof

According to (14.3), (10.2), (10.3), we get $\mathring{\mathbb {V}}_{0}\left( {z}\right) = \left[ \begin{array}{cc}O_{{q\times q}}&{}-\mathfrak {k}_{0}\\ \left( {z-\alpha }\right) \mathfrak {k}_{0}^{\mathord {+}}&{}\left( {z-\alpha }\right) I_{q}\end{array}\right]$. Taking additionally into account $\mathfrak {k}_{0}=s_{0}$, then (16.3) follows. $\square$

Proposition 16.4

(cf. [19, Lem. 14.11]) Let $s_{0}\in \mathbb {C}_\succcurlyeq ^{{q\times q}}$, let $w\in \Pi _{\mathord {+}}$, and let $X\in \mathscr {K}_{0}\left( {w}\right) \cap \mathscr {S}_{q,\alpha }\left( {w}\right)$. Then there exists a function $F\in {\mathcal {S}_{0,q}\left[ {\mathbb {C}\backslash {[\alpha ,\infty )};(s_j)_{j=0}^{0},\preccurlyeq } \right] }$ fulfilling $F\left( {w}\right) =X$.

Proof

Our proof contains some arguments which are also used in [19, Lem. 14.11]. According to Corollary 7.4 there exists a sequence $(s_{j})_{j=1}^{\infty }$ of complex ${q\times q}$ matrices such that $(s_j)_{j=0}^{\infty }\in \mathcal {K}^\succcurlyeq _{q,\infty ,\alpha }=\mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,\infty ,\alpha }$. Consequently, from Remark 7.6 we can infer $(s_j)_{j=0}^{\infty }\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,\infty }=\mathcal {H}^\succcurlyeq _{q,\infty }$. Let $(\mathfrak {k}_{j})_{j=0}^{\infty }$ be the $\mathcal {K}_\alpha$-parameter sequence of $(s_j)_{j=0}^{\infty }$ and let $(\mathfrak {h}_{j})_{j=0}^{\infty }$ be the $\mathcal {H}$-parameter sequence of $(s_j)_{j=0}^{\infty }$. According to Remark 9.2, we have then

$$\begin{aligned} \mathfrak {k}_{2k}=\mathfrak {h}_{2k}\quad \text {for all }k \in \mathbb {N}_0. \end{aligned}$$

(16.4)

Remark 9.3 shows (14.20). Since $w\in \Pi _{\mathord {+}}$, we have $\Im w\in (0,\infty )$. Recall that $\mathbb {K}_{{q\times q}}$ stands for the set of all contractive complex ${q\times q}$ matrices. For the sake of improved readability, from hereon our proof is divided into six parts.

Part 1: Due to $X\in \mathscr {K}_{0}\left( {w}\right) \cap \mathscr {S}_{q,\alpha }\left( {w}\right)$, we have, in particular, $X\in \mathscr {K}_{0}\left( {w}\right)$. Hence, according to (13.1) and Notation 6.7, there exists a matrix $C\in \mathbb {K}_{{q\times q}}$ fulfilling

$$\begin{aligned} X =\mathscr {C}_{0}\left( {w}\right) +\left( {w-\overline{w}}\right) ^{-1}\mathscr {A}_{0}\left( {w}\right) C\mathscr {B}_{0}\left( {w}\right) . \end{aligned}$$

(16.5)

Regarding $(s_j)_{j=0}^{\infty }\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,\infty }$, let $(\chi _{j})_{j=-1}^{\infty }$ be the sequence of $\chi$-functions associated with $(s_j)_{j=0}^{\infty }$. Let $E:=-\left[ {\chi _{1}\left( {w}\right) } \right] ^*$, let $B:=\left( {\Im w}\right) ^{-1}\Im E$, let $P:=\mathbb {P}_{\mathcal {R}\left( {E}\right) }$, and let $Q:=\mathbb {P}_{\mathcal {N}\left( {E}\right) }$. Clearly, then

$$\begin{aligned} \chi _{1}\left( {w}\right)&=-E^*,&\left[ {\chi _{1}\left( {w}\right) } \right] ^*&=-E,{} & {} \text {and}&\left( {\Im w}\right) ^{-1}\Im \chi _{1}\left( {w}\right)&=B. \end{aligned}$$

(16.6)

From Remark 6.3 we can thus infer $B\in \mathbb {C}_\succcurlyeq ^{{q\times q}}$. Obviously, the (constant) matrix-valued functions $\phi ,\psi :\Pi _{\mathord {+}}\rightarrow \mathbb {C}^{{q\times q}}$ defined by

$$\begin{aligned} \phi \left( {z}\right)&:=E\sqrt{B}^{\mathord {+}}-E^*\sqrt{B}^{\mathord {+}}CP{} & {} \text {and}&\psi \left( {z}\right)&:=\sqrt{B}^{\mathord {+}}-\sqrt{B}^{\mathord {+}}CP+Q, \end{aligned}$$

(16.7)

respectively, fulfill $\mathbb {H}\left( {\phi }\right) =\mathbb {H}\left( {\psi }\right) =\Pi _{\mathord {+}}$. Proposition 6.4 yields $\mathcal {R}\left( {\Im \chi _{1}\left( {z}\right) }\right) =\mathcal {R}\left( {\left[ {\chi _{1}\left( {z}\right) } \right] ^*}\right) =\mathcal {R}\left( {\mathfrak {h}_{0}}\right)$ and $\mathcal {N}\left( {\Im \chi _{1}\left( {z}\right) }\right) =\mathcal {N}\left( {\left[ {\chi _{1}\left( {z}\right) } \right] ^*}\right) =\mathcal {N}\left( {\mathfrak {h}_{0}}\right)$ for all $z\in \Pi _{\mathord {+}}$. By virtue of (16.6), consequently,

$$\begin{aligned} \mathcal {R}\left( {B}\right) =\mathcal {R}\left( {E}\right) =\mathcal {R}\left( {\mathfrak {h}_{0}}\right) \quad \text {and}\quad \mathcal {N}\left( {B}\right) =\mathcal {N}\left( {E}\right) =\mathcal {N}\left( {\mathfrak {h}_{0}}\right) \end{aligned}$$

(16.8)

follow. Regarding $\Im w\in (0,\infty )$ and taking into account $B\in \mathbb {C}_\succcurlyeq ^{{q\times q}}$ and $\mathcal {R}\left( {B}\right) =\mathcal {R}\left( {E}\right)$ as well as (16.7) and $C\in \mathbb {K}_{{q\times q}}$, we can apply Lemma A.16 to discern that

$$\begin{aligned} {{\,\textrm{rank}\,}}\begin{bmatrix}\phi \left( {z}\right) \\ \psi \left( {z}\right) \end{bmatrix}&=q \quad \text {and}\quad \Im \left( {\left[ {\psi \left( {z}\right) } \right] ^*\phi \left( {z}\right) }\right) \in \mathbb {C}_\succcurlyeq ^{{q\times q}}, \end{aligned}$$

(16.9)

that

$$\begin{aligned} P\phi \left( {z}\right)&=\phi \left( {z}\right) ,&\phi \left( {z}\right) P&=\phi \left( {z}\right) ,&\psi \left( {z}\right) P&=\psi \left( {z}\right) -Q, \end{aligned}$$

(16.10)

and

$$\begin{aligned} \sqrt{B}^{\mathord {+}}\left[ {\phi \left( {z}\right) -E\psi \left( {z}\right) } \right] \left[ {\phi \left( {z}\right) -E^*\psi \left( {z}\right) } \right] ^{\mathord {+}}\sqrt{B} =PCP\end{aligned}$$

(16.11)

hold true for all $z\in \Pi _{\mathord {+}}$. In view of Definition 4.2 and Remark 4.3, from $\mathbb {H}\left( {\phi }\right) =\mathbb {H}\left( {\psi }\right) =\Pi _{\mathord {+}}$ and (16.9) we recognize that the pair $\left( {\phi };{\psi }\right)$ belongs to $\mathcal{P}\mathcal{R}_{q}\left( {\Pi _{\mathord {+}}}\right)$ and that the set $\mathcal {D}:=\emptyset$ belongs to $\mathscr {D}\left( {\phi ,\psi }\right)$. Regarding (16.8), we have

$$\begin{aligned} P&=\mathbb {P}_{\mathcal {R}\left( {E}\right) }=\mathbb {P}_{\mathcal {R}\left( {\mathfrak {h}_{0}}\right) }\quad \text {and}\quad Q=\mathbb {P}_{\mathcal {N}\left( {E}\right) }=\mathbb {P}_{\mathcal {N}\left( {\mathfrak {h}_{0}}\right) }. \end{aligned}$$

(16.12)

From (16.12) and (16.10) we receive $\mathbb {P}_{\mathcal {R}\left( {\mathfrak {h}_{0}}\right) }\phi \left( {z}\right) =\phi \left( {z}\right)$ for all $z\in \Pi _{\mathord {+}}$. According to Notation 4.5, hence $\left( {\phi };{\psi }\right) \in \mathcal {P}\left[ {\mathfrak {h}_{0}} \right]$. Let $\left[ {(\mathfrak {a}_{k})_{k=0}^{\infty },(\mathfrak {b}_{k})_{k=0}^{\infty },(\mathfrak {c}_{k})_{k=0}^{\infty },(\mathfrak {d}_{k})_{k=0}^{\infty }} \right]$ be the $\mathbb {R}$-QMP associated with $(s_j)_{j=0}^{\infty }$ and let $\left[ {(\textbf{a}_{k})_{k=0}^{\infty },(\textbf{b}_{k})_{k=0}^{\infty },(\textbf{c}_{k})_{k=0}^{\infty },(\textbf{d}_{k})_{k=0}^{\infty }} \right]$ be the first ${[\alpha ,\infty )}$-QMP associated with $(s_j)_{j=0}^{\infty }$. Regarding $\mathcal {D}\in \mathscr {D}\left( {\phi ,\psi }\right)$ as well as $\mathcal {D}=\emptyset$, we can infer from Corollary 11.14 then that

$$\begin{aligned} \det \left( {\mathfrak {b}_{0}\left( {z}\right) \mathfrak {h}_{0}^{\mathord {+}}\phi \left( {z}\right) +\mathfrak {b}_{1}\left( {z}\right) \psi \left( {z}\right) }\right)&\ne 0,&\det \left( {\textbf{b}_{0}\left( {z}\right) \mathfrak {h}_{0}^{\mathord {+}}\phi \left( {z}\right) +\textbf{b}_{1}\left( {z}\right) \psi \left( {z}\right) }\right)&\ne 0, \end{aligned}$$

(16.13)

and

$$\begin{aligned}{} & {} -\left[ {\mathfrak {a}_{0}\left( {z}\right) \mathfrak {h}_{0}^{\mathord {+}}\phi \left( {z}\right) +\mathfrak {a}_{1}\left( {z}\right) \psi \left( {z}\right) } \right] \left[ {\mathfrak {b}_{0}\left( {z}\right) \mathfrak {h}_{0}^{\mathord {+}}\phi \left( {z}\right) +\mathfrak {b}_{1}\left( {z}\right) \psi \left( {z}\right) } \right] ^{-1}\nonumber \\{} & {} \quad =-\left[ {\textbf{a}_{0}\left( {z}\right) \mathfrak {h}_{0}^{\mathord {+}}\phi \left( {z}\right) +\textbf{a}_{1}\left( {z}\right) \psi \left( {z}\right) } \right] \left[ {\textbf{b}_{0}\left( {z}\right) \mathfrak {h}_{0}^{\mathord {+}}\phi \left( {z}\right) +\textbf{b}_{1}\left( {z}\right) \psi \left( {z}\right) } \right] ^{-1}\qquad \end{aligned}$$

(16.14)

hold true for all $z\in \Pi _{\mathord {+}}$. Setting

$$\begin{aligned} S:=\phi \left( {w}\right) \quad \text {and}\quad T:=\psi \left( {w}\right) \end{aligned}$$

(16.15)

we see, in view of (16.13) and (16.14), that

$$\begin{aligned} {\mathfrak {R}}&:=\mathfrak {b}_{0}\left( {w}\right) \mathfrak {h}_{0}^{\mathord {+}}S+\mathfrak {b}_{1}\left( {w}\right) T{} & {} \text {and}&{\textbf{R}}&:=\textbf{b}_{0}\left( {w}\right) \mathfrak {h}_{0}^{\mathord {+}}S+\textbf{b}_{1}\left( {w}\right) T\end{aligned}$$

(16.16)

satisfy $\det {\mathfrak {R}}\ne 0$ and $\det {\textbf{R}}\ne 0$ as well as

$$\begin{aligned} -\left[ {\mathfrak {a}_{0}\left( {w}\right) \mathfrak {h}_{0}^{\mathord {+}}S+\mathfrak {a}_{1}\left( {w}\right) T} \right] {\mathfrak {R}}^{-1}=-\left[ {\textbf{a}_{0}\left( {w}\right) \mathfrak {h}_{0}^{\mathord {+}}S+\textbf{a}_{1}\left( {w}\right) T} \right] {\textbf{R}}^{-1}. \end{aligned}$$

(16.17)

Now we are going to justify that all assumptions for the application of Proposition 6.11 to the sequence $(s_j)_{j=0}^{1}$ are satisfied. Because of $(s_j)_{j=0}^{\infty }\in \mathcal {H}^\succcurlyeq _{q,\infty }$, we have $(s_j)_{j=0}^{1}\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,1}$. According to (16.12), we have $P=\mathbb {P}_{\mathcal {R}\left( {\mathfrak {h}_{0}}\right) }$ and $Q=\mathbb {P}_{\mathcal {N}\left( {\mathfrak {h}_{0}}\right) }$. In view of $\Im w\in (0,\infty )$ and (16.15), from (16.9) and (16.10), we see $\Im \left( {w}\right) \Im \left( {T^*S}\right) \in \mathbb {C}_\succcurlyeq ^{{q\times q}}$ as well as

$$\begin{aligned} PS&=S,&SP&=S,{} & {} \text {and}&TP&=T-Q. \end{aligned}$$

(16.18)

Using additionally (16.16) and $\det {\mathfrak {R}}\ne 0$, we can thus apply Proposition 6.11 to infer that

$$\begin{aligned} {\textbf{K}} :=\sqrt{\left( {\Im w}\right) ^{-1}\Im \chi _{1}\left( {w}\right) }^{\mathord {+}}\left( {S+\left[ {\chi _{1}\left( {w}\right) } \right] ^*T}\right) \left( {S+\chi _{1}\left( {w}\right) T}\right) ^{\mathord {+}}\sqrt{\left( {\Im w}\right) ^{-1}\Im \chi _{1}\left( {w}\right) } \end{aligned}$$

(16.19)

is a contractive matrix which fulfills

$$\begin{aligned} -\left[ {\mathfrak {a}_{0}\left( {w}\right) \mathfrak {h}_{0}^{\mathord {+}}S+\mathfrak {a}_{1}\left( {w}\right) T} \right] {\mathfrak {R}}^{-1}=\mathscr {C}_{1}\left( {w}\right) +\left( {w-\overline{w}}\right) ^{-1}\mathscr {A}_{1}\left( {w}\right) {\textbf{K}}\mathscr {B}_{1}\left( {w}\right) . \end{aligned}$$

(16.20)

By virtue of (16.19), (16.6), (16.15), and (16.11), we discern

$$\begin{aligned} {\textbf{K}} =\sqrt{B}^{\mathord {+}}\left[ {\phi \left( {w}\right) -E\psi \left( {w}\right) } \right] \left[ {\phi \left( {w}\right) -E^*\psi \left( {w}\right) } \right] ^{\mathord {+}}\sqrt{B} =PCP. \end{aligned}$$

(16.21)

The application of Lemma 6.9 to the sequence $(s_j)_{j=0}^{\infty }\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,\infty }$ yields

$$\begin{aligned} \mathscr {A}_{0}\left( {w}\right)&=\mathscr {A}_{1}\left( {w}\right) ,&\mathscr {B}_{0}\left( {w}\right)&=\mathscr {B}_{1}\left( {w}\right) ,{} & {} \text {and}&\mathscr {C}_{0}\left( {w}\right)&=\mathscr {C}_{1}\left( {w}\right) . \end{aligned}$$

(16.22)

From Proposition 6.10 we get $\mathcal {N}\left( {\mathscr {A}_{0}\left( {w}\right) }\right) =\mathcal {N}\left( {\mathfrak {h}_{0}}\right)$ and $\mathcal {R}\left( {\mathscr {B}_{0}\left( {w}\right) }\right) =\mathcal {R}\left( {\mathfrak {h}_{0}}\right)$. According to (16.12), then $P\mathscr {B}_{0}\left( {w}\right) =\mathscr {B}_{0}\left( {w}\right)$. By virtue of (16.12) and Remark A.8, we see $\mathcal {R}\left( {P}\right) =\mathcal {R}\left( {\mathfrak {h}_{0}}\right)$ and $\mathcal {R}\left( {Q}\right) =\mathcal {N}\left( {\mathfrak {h}_{0}}\right)$. Hence, $\mathscr {A}_{0}\left( {w}\right) Q=O_{{q\times q}}$. Remark A.7 provides $\mathcal {N}\left( {\mathfrak {h}_{0}^*}\right) =\mathcal {R}\left( {\mathfrak {h}_{0}}\right) ^\bot$. Since, in view of $(s_j)_{j=0}^{\infty }\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,\infty }$, Remark 5.3 yields $\mathfrak {h}_{0}^*=\mathfrak {h}_{0}$, then $\mathcal {N}\left( {\mathfrak {h}_{0}}\right) =\mathcal {R}\left( {\mathfrak {h}_{0}}\right) ^\bot$ follows. Thus, Remark A.9 yields $P+Q=I_{q}$, implying $\mathscr {A}_{0}\left( {w}\right) P=\mathscr {A}_{0}\left( {w}\right)$. Consequently, we infer $\mathscr {A}_{0}\left( {w}\right) PCP\mathscr {B}_{0}\left( {w}\right) =\mathscr {A}_{0}\left( {w}\right) C\mathscr {B}_{0}\left( {w}\right)$. Applying additionally (16.5), (16.22), (16.21), (16.20), and (16.17), we then conclude

$$\begin{aligned}\begin{aligned} X&=\mathscr {C}_{0}\left( {w}\right) +\left( {w-\overline{w}}\right) ^{-1}\mathscr {A}_{0}\left( {w}\right) C\mathscr {B}_{0}\left( {w}\right) \\&=\mathscr {C}_{0}\left( {w}\right) +\left( {w-\overline{w}}\right) ^{-1}\mathscr {A}_{0}\left( {w}\right) PCP\mathscr {B}_{0}\left( {w}\right) \\&=\mathscr {C}_{1}\left( {w}\right) +\left( {w-\overline{w}}\right) ^{-1}\mathscr {A}_{1}\left( {w}\right) {\textbf{K}}\mathscr {B}_{1}\left( {w}\right) \\&=-\left[ {\mathfrak {a}_{0}\left( {w}\right) \mathfrak {h}_{0}^{\mathord {+}}S+\mathfrak {a}_{1}\left( {w}\right) T} \right] {\mathfrak {R}}^{-1}=-\left[ {\textbf{a}_{0}\left( {w}\right) \mathfrak {h}_{0}^{\mathord {+}}S+\textbf{a}_{1}\left( {w}\right) T} \right] {\textbf{R}}^{-1}. \end{aligned}\end{aligned}$$

Using additionally $\det {\textbf{R}} \ne 0$, (16.16), (16.4), and Lemma 14.2(a), we then obtain

$$\begin{aligned} \begin{aligned} \begin{bmatrix} X\\ I_{q}\end{bmatrix}&=\begin{bmatrix} -\left[ {\textbf{a}_{0}\left( {w}\right) \mathfrak {h}_{0}^{\mathord {+}}S+\textbf{a}_{1}\left( {w}\right) T} \right] {\textbf{R}}^{-1}\\ {\textbf{R}} {\textbf{R}}^{-1}\end{bmatrix} =\begin{bmatrix} -\textbf{a}_{0}\left( {w}\right) \mathfrak {h}_{0}^{\mathord {+}}S-\textbf{a}_{1}\left( {w}\right) T\\ \textbf{b}_{0}\left( {w}\right) \mathfrak {h}_{0}^{\mathord {+}}S+\textbf{b}_{1}\left( {w}\right) T\end{bmatrix} {\textbf{R}}^{-1}\\&=\begin{bmatrix} -\textbf{a}_{0}\left( {w}\right) \mathfrak {k}_{0}^{\mathord {+}}&{}-\textbf{a}_{1}\left( {w}\right) \\ \textbf{b}_{0}\left( {w}\right) \mathfrak {k}_{0}^{\mathord {+}}&{}\textbf{b}_{1}\left( {w}\right) \end{bmatrix}\begin{bmatrix} S\\ T\end{bmatrix}{\textbf{R}}^{-1}=\mathfrak {V}^{\left( {\alpha }\right) }_{1}\left( {w}\right) \begin{bmatrix} S\\ T\end{bmatrix}{\textbf{R}}^{-1}. \end{aligned}\end{aligned}$$

(16.23)

Part 2: Regarding the assumption $X\in \mathscr {K}_{0}\left( {w}\right) \cap \mathscr {S}_{q,\alpha }\left( {w}\right)$, in particular, $X\in \mathscr {S}_{q,\alpha }\left( {w}\right)$. Then, according to (16.2), we have $\Im \left( {\eta X}\right) \in \mathbb {C}_\succcurlyeq ^{{q\times q}}$, where $\eta :=w-\alpha$. Let $S_{\mathord {\circ }}:=\eta X+\mathfrak {k}_{0}$ and $T_{\mathord {\circ }}:=I_{q}$. Using Remark A.1 and taking into account that (14.20) implies $\Im \mathfrak {k}_{0}=O_{{q\times q}}$, we get

$$\begin{aligned} \Im \left( {T_{\mathord {\circ }}^*S_{\mathord {\circ }}}\right) =\Im \left( {\eta X+\mathfrak {k}_{0}}\right) =\Im \left( {\eta X}\right) +\Im \mathfrak {k}_{0} =\Im \left( {\eta X}\right) \in \mathbb {C}_\succcurlyeq ^{{q\times q}}. \end{aligned}$$

(16.24)

Regarding $\mathfrak {k}_{0}=s_{0}$, we obtain furthermore

$$\begin{aligned} \begin{bmatrix} \eta I_{q}&{}s_{0}\\ O_{{q\times q}}&{}I_{q}\end{bmatrix} \begin{bmatrix}X\\ I_{q}\end{bmatrix} = \begin{bmatrix} S_{\mathord {\circ }}\\ T_{\mathord {\circ }}\end{bmatrix}. \end{aligned}$$

(16.25)

Part 3: Obviously, the matrix ${\mathbb {D}}:=\bigl [{\begin{matrix}\eta I_{q}&{}\mathfrak {k}_{1}\\ O_{{q\times q}}&{}I_{q}\end{matrix}}\bigr ]$ is invertible with

(16.26)

From (16.15) and (16.9), we see ${{\,\textrm{rank}\,}}\bigl [{\begin{matrix}S\\ T\end{matrix}}\bigr ]=q$. Consequently, the ${q\times q}$ block representation

$$\begin{aligned} {\mathbb {D}}^{-1}\begin{bmatrix} S\\ T\end{bmatrix}{\textbf{R}}^{-1}=\begin{bmatrix}Y\\ Z\end{bmatrix} \end{aligned}$$

(16.27)

of ${\mathbb {D}}^{-1}\bigl [{\begin{matrix}S\\ T\end{matrix}}\bigr ]{\textbf{R}}^{-1}$ fulfills ${{\,\textrm{rank}\,}}\bigl [{\begin{matrix}Y\\ Z\end{matrix}}\bigr ]={{\,\textrm{rank}\,}}\bigl [{\begin{matrix}S\\ T\end{matrix}}\bigr ]=q$. Lemma 14.5 shows $\mathfrak {V}^{\left( {\alpha }\right) }_{1}\left( {w}\right) {\mathbb {D}}=\mathring{\mathbb {V}}_{0}\left( {w}\right)$. Hence, using additionally (16.27) and (16.23), we deduce

$$\begin{aligned} \mathring{\mathbb {V}}_{0}\left( {w}\right) \begin{bmatrix}Y\\ Z\end{bmatrix} =\mathfrak {V}^{\left( {\alpha }\right) }_{1}\left( {w}\right) \begin{bmatrix} S\\ T\end{bmatrix}{\textbf{R}}^{-1}=\begin{bmatrix}X\\ I_{q}\end{bmatrix}. \end{aligned}$$

(16.28)

In view of (16.12), (16.4), and Remark A.10, we can infer

$$\begin{aligned} P=\mathbb {P}_{\mathcal {R}\left( {\mathfrak {k}_{0}}\right) } =\mathfrak {k}_{0}\mathfrak {k}_{0}^{\mathord {+}}. \end{aligned}$$

(16.29)

Lemma 16.3 then shows $\left[ \begin{array}{cc}\eta I_{q}&{}s_{0}\\ O_{{q\times q}}&{}I_{q}\end{array}\right] \mathring{\mathbb {V}}_{0}\left( {w}\right) =\eta \left[ \begin{array}{cc}P&{}O_{{q\times q}}\\ \mathfrak {k}_{0}^{\mathord {+}}&{}I_{q}\end{array}\right]$. Consequently, applying additionally (16.25) and (16.28), we conclude

$$\begin{aligned} \begin{bmatrix}S_{\mathord {\circ }}\\ T_{\mathord {\circ }}\end{bmatrix} = \begin{bmatrix} \eta I_{q}&{}s_{0}\\ O_{{q\times q}}&{}I_{q}\end{bmatrix} \mathring{\mathbb {V}}_{0}\left( {w}\right) \begin{bmatrix}Y\\ Z\end{bmatrix} =\eta \begin{bmatrix}P&{}O_{{q\times q}}\\ \mathfrak {k}_{0}^{\mathord {+}}&{}I_{q}\end{bmatrix} \begin{bmatrix}Y\\ Z\end{bmatrix}. \end{aligned}$$

(16.30)

Part 4: Regarding (16.27) and (16.26), we have

$$\begin{aligned} Y =\eta ^{-1}\left( { S-\mathfrak {k}_{1} T}\right) {\textbf{R}}^{-1}\quad \text {and}\quad Z=T{\textbf{R}}^{-1}. \end{aligned}$$

(16.31)

In view of $(s_j)_{j=0}^{\infty }\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,\infty ,\alpha }$, Remark 9.3 yields $\mathfrak {k}_{0}\mathfrak {k}_{0}^{\mathord {+}}\mathfrak {k}_{1}=\mathfrak {k}_{1}$. By virtue of (16.29), then $P\mathfrak {k}_{1}=\mathfrak {k}_{1}$ follows. Using additionally (16.18), from the first identity in (16.31) we then obtain

$$\begin{aligned} \begin{aligned} PY =\eta ^{-1}\left( {PS-P\mathfrak {k}_{1}T}\right) {\textbf{R}}^{-1}=\eta ^{-1}\left( {S-\mathfrak {k}_{1}T}\right) {\textbf{R}}^{-1}=Y. \end{aligned}\end{aligned}$$

(16.32)

Regarding (16.30), we have $S_{\mathord {\circ }}=\eta PY$ and $T_{\mathord {\circ }}=\eta \left( {\mathfrak {k}_{0}^{\mathord {+}}Y+Z}\right)$. In view of (14.20), we infer from Remark A.12 that $\left( {\mathfrak {k}_{0}^{\mathord {+}}}\right) ^*=\mathfrak {k}_{0}^{\mathord {+}}$. Taking additionally into account (16.32), we obtain $T_{\mathord {\circ }}^*S_{\mathord {\circ }}=|{\eta } |^2\left( {Y^*\mathfrak {k}_{0}^{\mathord {+}}Y+Z^*Y}\right)$ and, thus, we conclude $Z^*Y=|{\eta } |^{-2}T_{\mathord {\circ }}^*S_{\mathord {\circ }}-Y^*\mathfrak {k}_{0}^{\mathord {+}}Y$. Using Remark A.1 and taking into account (16.24) and that $\left( {\mathfrak {k}_{0}^{\mathord {+}}}\right) ^*=\mathfrak {k}_{0}^{\mathord {+}}$ implies $\Im \left( {Y^*\mathfrak {k}_{0}^{\mathord {+}}Y}\right) =O_{{q\times q}}$, we then infer

$$\begin{aligned} \Im \left( {Z^*Y}\right) =|{\eta } |^{-2}\Im \left( {T_{\mathord {\circ }}^*S_{\mathord {\circ }}}\right) -\Im \left( {Y^*\mathfrak {k}_{0}^{\mathord {+}}Y}\right) =|{\eta } |^{-2}\Im \left( { T_{\mathord {\circ }}^*S_{\mathord {\circ }}}\right) \in \mathbb {C}_\succcurlyeq ^{{q\times q}}. \end{aligned}$$

(16.33)

Because of (14.20), we have $\Im \left( {T^*\mathfrak {k}_{1}T}\right) =O_{{q\times q}}$. From (16.15) and (16.9) we see $\Im \left( {T^*S}\right) \in \mathbb {C}_\succcurlyeq ^{{q\times q}}$. Hence, using (16.31) as well as Remarks A.2, A.1, and A.6, we infer

$$\begin{aligned}{} & {} \Im \left( {\eta Z^*Y}\right) =\Im \left( {\eta \left( { T{\textbf{R}}^{-1}}\right) ^*\left[ {\eta ^{-1}\left( { S-\mathfrak {k}_{1} T}\right) {\textbf{R}}^{-1}} \right] }\right) \nonumber \\{} & {} \quad =\Im \left( {{\textbf{R}}^{-*}T^*\left( { S-\mathfrak {k}_{1} T}\right) {\textbf{R}}^{-1}}\right) ={\textbf{R}}^{-*}\left[ {\Im \left( { T^*S}\right) -\Im \left( { T^*\mathfrak {k}_{1} T}\right) } \right] {\textbf{R}}^{-1}\nonumber \\{} & {} \quad ={\textbf{R}}^{-*}\left[ {\Im \left( { T^*S}\right) } \right] {\textbf{R}}^{-1}\in \mathbb {C}_\succcurlyeq ^{{q\times q}}. \end{aligned}$$

(16.34)

Part 5: Let $\pi ,\rho :\mathbb {C}\rightarrow \mathbb {C}^{{q\times q}}$ be defined by

$$\begin{aligned} \pi \left( {z}\right) :=Y \,\text {and}\, \rho \left( {z}\right) :=\frac{w-z}{\Im w}HY+Z, \,\text {where}\, H:=\left( {Y^{\mathord {+}}}\right) ^*\left[ {\Im \left( {Z^*Y}\right) } \right] Y^{\mathord {+}}. \nonumber \\ \end{aligned}$$

(16.35)

In view of (16.33) and (16.35), we can apply Lemma 8.5 and it follows that $\pi$ and $\rho$ are both holomorphic in $\mathbb {C}$ fulfilling

$$\begin{aligned} \begin{bmatrix}\pi \left( {w}\right) \\ \rho \left( {w}\right) \end{bmatrix} =\begin{bmatrix}Y\\ Z\end{bmatrix} \end{aligned}$$

(16.36)

as well as

$$\begin{aligned} {{\,\textrm{rank}\,}}\begin{bmatrix}\pi \left( {z}\right) \\ \rho \left( {z}\right) \end{bmatrix}&={{\,\textrm{rank}\,}}\begin{bmatrix}Y\\ Z\end{bmatrix}, \end{aligned}$$

(16.37)

$$\begin{aligned} \Im \left( {\left[ {\rho \left( {z}\right) } \right] ^*\pi \left( {z}\right) }\right)&=\frac{\Im z}{\Im w}\Im \left( {Z^*Y}\right) , \end{aligned}$$

(16.38)

and

$$\begin{aligned} \Im \left( {\left( {z-\alpha }\right) \left[ {\rho \left( {z}\right) } \right] ^*\pi \left( {z}\right) }\right)&=\frac{\Im z}{\Im w}\Im \left( {\eta Z^*Y}\right) \end{aligned}$$

(16.39)

for all $z\in \mathbb {C}$. Let $\phi _{\mathord {\bullet }},\psi _{\mathord {\bullet }}:\mathbb {C}\backslash {[\alpha ,\infty )}\rightarrow \mathbb {C}^{{q\times q}}$ be defined by

$$\begin{aligned} \phi _{\mathord {\bullet }}\left( {z}\right) :=\pi \left( {z}\right) \quad \text {and}\quad \psi _{\mathord {\bullet }}\left( {z}\right) :=\rho \left( {z}\right) . \end{aligned}$$

(16.40)

Clearly, $\mathcal {D}_{\mathord {\bullet }}:=\emptyset$ is a discrete subset of $\mathbb {C}\backslash {[\alpha ,\infty )}$. Keeping in mind that $\pi$ and $\rho$ are holomorphic in $\mathbb {C}$, we see that $\phi _{\mathord {\bullet }}$ and $\psi _{\mathord {\bullet }}$ are holomorphic and, in particular, meromorphic in $\mathbb {C}\backslash {[\alpha ,\infty )}$. For all $z\in \mathbb {C}\backslash {[\alpha ,\infty )}$, equations (16.40), (16.37), and ${{\,\textrm{rank}\,}}\bigl [{\begin{matrix}Y\\ Z\end{matrix}}\bigr ]=q$ imply ${{\,\textrm{rank}\,}}\bigl [{\begin{matrix}\phi _{\mathord {\bullet }}\left( {z}\right) \\ \psi _{\mathord {\bullet }}\left( {z}\right) \end{matrix}}\bigr ]={{\,\textrm{rank}\,}}\bigl [{\begin{matrix}\pi \left( {z}\right) \\ \rho \left( {z}\right) \end{matrix}}\bigr ]={{\,\textrm{rank}\,}}\bigl [{\begin{matrix}Y\\ Z\end{matrix}}\bigr ]=q$. Regarding $\Im w\in (0,\infty )$, for all $z\in \mathbb {C}\backslash \mathbb {R}$, moreover (16.40), (16.38), and (16.33) yield

$$\begin{aligned} \left( {\Im z}\right) ^{-1}\Im \left( {\left[ {\psi _{\mathord {\bullet }}\left( {z}\right) } \right] ^*\phi _{\mathord {\bullet }}\left( {z}\right) }\right) =\left( {\Im z}\right) ^{-1}\Im \left( {\left[ {\rho \left( {z}\right) } \right] ^*\pi \left( {z}\right) }\right) =\left( {\Im w}\right) ^{-1}\Im \left( {Z^*Y}\right) \in \mathbb {C}_\succcurlyeq ^{{q\times q}}, \end{aligned}$$

whereas (16.40), (16.39), and (16.34) provide

$$\begin{aligned}\begin{aligned} \left( {\Im z}\right) ^{-1}\Im \left( {\left( {z-\alpha }\right) \left[ {\psi _{\mathord {\bullet }}\left( {z}\right) } \right] ^*\phi _{\mathord {\bullet }}\left( {z}\right) }\right)&=\left( {\Im z}\right) ^{-1}\Im \left( {\left( {z-\alpha }\right) \left[ {\rho \left( {z}\right) } \right] ^*\pi \left( {z}\right) }\right) \\&=\left( {\Im w}\right) ^{-1}\Im \left( {\eta Z^*Y}\right) \in \mathbb {C}_\succcurlyeq ^{{q\times q}}. \end{aligned}\end{aligned}$$

In view of Remark 8.2, then, according to Definition 8.1, the pair $\left( {\phi _{\mathord {\bullet }}};{\psi _{\mathord {\bullet }}}\right)$ belongs to $\mathcal{P}\mathcal{S}_{q}\left( {\mathbb {C}\backslash {[\alpha ,\infty )}}\right)$ and the set $\mathcal {D}_{\mathord {\bullet }}$ belongs to $\mathscr {D}_{\mathord {\bullet }}\left( {\phi _{\mathord {\bullet }},\psi _{\mathord {\bullet }}}\right)$. Because of Definition 5.2 and (5.2), we have $\mathfrak {h}_{0}=L_{0}$. Using additionally (16.40), (16.12), (16.35), and (16.32), we can conclude $\mathbb {P}_{\mathcal {R}\left( {L_{0}}\right) }\phi _{\mathord {\bullet }}\left( {z}\right) =\mathbb {P}_{\mathcal {R}\left( {\mathfrak {h}_{0}}\right) }\pi \left( {z}\right) =PY=Y=\pi \left( {z}\right) =\phi _{\mathord {\bullet }}\left( {z}\right)$ for all $z\in \mathbb {C}\backslash {[\alpha ,\infty )}$. By virtue of Notation 8.4, consequently $\left( {\phi _{\mathord {\bullet }}};{\psi _{\mathord {\bullet }}}\right) \in \mathcal {P}_{{\mathord {\bullet }},\alpha }\left[ {L_{0}} \right]$.

Part 6: From $\left( {\phi _{\mathord {\bullet }}};{\psi _{\mathord {\bullet }}}\right) \in \mathcal {P}_{{\mathord {\bullet }},\alpha }\left[ {L_{0}} \right]$, Theorem 10.17(a) and the notations therein, we see that $\det \left( {\tilde{\textbf{p}}_{0}^\flat L_{0}^{\mathord {+}}\phi _{\mathord {\bullet }}+\tilde{\textbf{p}}_{1}\psi _{\mathord {\bullet }}}\right)$ does not vanish identically and that

$$\begin{aligned} F :=-\left( {\tilde{\textbf{q}}_{0}^\flat L_{0}^{\mathord {+}}\phi _{\mathord {\bullet }}+\tilde{\textbf{q}}_{1}\psi _{\mathord {\bullet }}}\right) \left( {\tilde{\textbf{p}}_{0}^\flat L_{0}^{\mathord {+}}\phi _{\mathord {\bullet }}+\tilde{\textbf{p}}_{1}\psi _{\mathord {\bullet }}}\right) ^{-1}\end{aligned}$$

(16.41)

belongs to ${\mathcal {S}_{0,q}\left[ {\mathbb {C}\backslash {[\alpha ,\infty )};(s_j)_{j=0}^{0},\preccurlyeq } \right] }$. Furthermore, regarding (16.28), (16.36), (16.40), and (14.3), we have

$$\begin{aligned}\begin{aligned} \begin{bmatrix}X\\ I_{q}\end{bmatrix}&=\mathring{\mathbb {V}}_{0}\left( {w}\right) \begin{bmatrix}Y\\ Z\end{bmatrix} =\mathring{\mathbb {V}}_{0}\left( {w}\right) \begin{bmatrix}\pi \left( {w}\right) \\ \rho \left( {w}\right) \end{bmatrix}\\&=\mathring{\mathbb {V}}_{0}\left( {w}\right) \begin{bmatrix}\phi _{\mathord {\bullet }}\left( {w}\right) \\ \psi _{\mathord {\bullet }}\left( {w}\right) \end{bmatrix} = \begin{bmatrix} -\eta \textbf{q}_{0}\left( {w}\right) \mathfrak {k}_{0}^{\mathord {+}}&{}-\textbf{q}_{1}\left( {w}\right) \\ \eta \textbf{p}_{0}\left( {w}\right) \mathfrak {k}_{0}^{\mathord {+}}&{}\textbf{p}_{1}\left( {w}\right) \end{bmatrix}\begin{bmatrix}\phi _{\mathord {\bullet }}\left( {w}\right) \\ \psi _{\mathord {\bullet }}\left( {w}\right) \end{bmatrix}. \end{aligned}\end{aligned}$$

Since $\mathfrak {k}_{0}=\mathfrak {h}_{0}=L_{0}$, we consequently get

$$\begin{aligned} X =-\eta \textbf{q}_{0}\left( {w}\right) \mathfrak {k}_{0}^{\mathord {+}}\phi _{\mathord {\bullet }}\left( {w}\right) -\textbf{q}_{1}\left( {w}\right) \psi _{\mathord {\bullet }}\left( {w}\right) =-\left[ {\tilde{\textbf{q}}_{0}^\flat \left( {w}\right) L_{0}^{\mathord {+}}\phi _{\mathord {\bullet }}\left( {w}\right) +\tilde{\textbf{q}}_{1}\left( {w}\right) \psi _{\mathord {\bullet }}\left( {w}\right) } \right] \end{aligned}$$

and

$$\begin{aligned} I_{q}=\eta \textbf{p}_{0}\left( {w}\right) \mathfrak {k}_{0}^{\mathord {+}}\phi _{\mathord {\bullet }}\left( {w}\right) +\textbf{p}_{1}\left( {w}\right) \psi _{\mathord {\bullet }}\left( {w}\right) =\tilde{\textbf{p}}_{0}^\flat \left( {w}\right) L_{0}^{\mathord {+}}\phi _{\mathord {\bullet }}\left( {w}\right) +\tilde{\textbf{p}}_{1}\left( {w}\right) \psi _{\mathord {\bullet }}\left( {w}\right) . \end{aligned}$$

In particular, $\det \left( {\tilde{\textbf{p}}_{0}^\flat \left( {w}\right) L_{0}^{\mathord {+}}\phi _{\mathord {\bullet }}\left( {w}\right) +\tilde{\textbf{p}}_{1}\left( {w}\right) \psi _{\mathord {\bullet }}\left( {w}\right) }\right) \ne 0$. Using additionally (16.41), we finally see $X=-\left[ {\tilde{\textbf{q}}_{0}^\flat \left( {w}\right) L_{0}^{\mathord {+}}\phi _{\mathord {\bullet }}\left( {w}\right) +\tilde{\textbf{q}}_{1}\left( {w}\right) \psi _{\mathord {\bullet }}\left( {w}\right) } \right] \left[ {\tilde{\textbf{p}}_{0}^\flat \left( {w}\right) L_{0}^{\mathord {+}}\phi _{\mathord {\bullet }}\left( {w}\right) +\tilde{\textbf{p}}_{1}\left( {w}\right) \psi _{\mathord {\bullet }}\left( {w}\right) } \right] ^{-1}=F\left( {w}\right)$. $\square$

We now get a description of the Weyl sets for the matricial Stieltjes moment problem where only the 0th moment $s_{0}$ is prescribed.

Theorem 16.5

(cf. [19, Satz 14.12]) If $s_{0}\in \mathbb {C}_\succcurlyeq ^{{q\times q}}$ and $w\in \Pi _{\mathord {+}}$, then

$$\begin{aligned} \left\{ {F\left( {w}\right) }:{F\in {\mathcal {S}_{0,q}\left[ {\mathbb {C}\backslash {[\alpha ,\infty )};(s_j)_{j=0}^{0},\preccurlyeq } \right] }}\right\} =\mathscr {K}_{0}\left( {w}\right) \cap \mathscr {S}_{q,\alpha }\left( {w}\right) . \end{aligned}$$

Proof

Combine Propositions 16.2 and 16.4. $\square$

Appendices

Appendix A: Some Facts on Matrix Theory

Remark A.1

If $A,B\in \mathbb {C}^{{q\times q}}$ and $\xi \in \mathbb {R}$, then $\Im \left( {A+B}\right) =\Im \left( {A}\right) +\Im \left( {B}\right)$ and $\Im \left( {\xi A}\right) =\xi \Im \left( {A}\right)$.

Remark A.2

If $A\in \mathbb {C}^{{q\times q}}$ and $B\in \mathbb {C}^{{p\times q}}$, then $\Im \left( {BAB^*}\right) =B\left( {\Im A}\right) B^*$.

Remark A.3

If $\zeta \in \mathbb {C}$ and $X\in \mathbb {C}_\textrm{H}^{{q\times q}}$, then $\Im \left( {\zeta X}\right) =\Im \left( {\zeta }\right) X$.

Remark A.4

If $\zeta \in \mathbb {C}$ and $M\in \mathbb {C}^{{q\times q}}$, then $\Im \left( {\zeta M }\right) =\Re \left( {\zeta }\right) \Im \left( {M}\right) +\Im \left( {\zeta }\right) \Re \left( {M}\right)$.

Remark A.5

If $M\in \mathbb {C}^{{q\times q}}$ satisfies $\Im M\in \mathbb {C}_\succcurlyeq ^{{q\times q}}$, then $\mathcal {N}\left( {M}\right) \subseteq \mathcal {N}\left( {\Im M}\right)$.

Remark A.6

If $A\in \mathbb {C}_\succcurlyeq ^{{q\times q}}$ and $B\in \mathbb {C}^{{q\times p}}$, then $B^*AB\in \mathbb {C}_\succcurlyeq ^{{p\times p}}$.

Remark A.7

If $A\in \mathbb {C}^{{p\times q}}$, then $\mathcal {R}\left( {A^*}\right) =\mathcal {N}\left( {A}\right) ^\bot$ and $\mathcal {N}\left( {A^*}\right) =\mathcal {R}\left( {A}\right) ^\bot$.

Remark A.8

Let $\mathcal {U}$ be a linear subspace of $\mathbb {C}^{q}$. Then $\mathbb {P}_{\mathcal {U}}$ is the unique complex ${q\times q}$ matrix satisfying $\mathbb {P}_{\mathcal {U}}^2=\mathbb {P}_{\mathcal {U}}$ and $\mathbb {P}_{\mathcal {U}}^*=\mathbb {P}_{\mathcal {U}}$ as well as $\mathcal {R}\left( {\mathbb {P}_{\mathcal {U}}}\right) =\mathcal {U}$.

Remark A.9

If $\mathcal {U}$ is a linear subspace of $\mathbb {C}^{q}$, then $\mathbb {P}_{\mathcal {U}}+\mathbb {P}_{\mathcal {U}^\bot }=I_{q}$.

Remark A.10

If $A\in \mathbb {C}^{{p\times q}}$, then $AA^{\mathord {+}}=\mathbb {P}_{\mathcal {R}\left( {A}\right) }$.

Lemma A.11

(cf. [12, Lem. A.13]) If $A,B\in \mathbb {C}_\textrm{H}^{{q\times q}}$ satisfy $O_{{q\times q}}\preccurlyeq A\preccurlyeq B$, then $O_{{q\times q}}\preccurlyeq \mathbb {P}_{\mathcal {R}\left( {A}\right) }B^{\mathord {+}}\mathbb {P}_{\mathcal {R}\left( {A}\right) }\preccurlyeq A^{\mathord {+}}$.

Remark A.12

If $A\in \mathbb {C}^{{p\times q}}$, then $\left( {A^{\mathord {+}}}\right) ^{\mathord {+}}=A$ and $\left( {A^*}\right) ^{\mathord {+}}=\left( {A^{\mathord {+}}}\right) ^*$.

Remark A.13

If $A\in \mathbb {C}^{{p\times q}}$, then $\mathcal {R}\left( {A^{\mathord {+}}}\right) =\mathcal {R}\left( {A^*}\right)$ and $\mathcal {N}\left( {A^{\mathord {+}}}\right) =\mathcal {N}\left( {A^*}\right)$.

Remark A.14

Let $A\in \mathbb {C}^{{p\times q}}$. In view of $AA^{\mathord {+}}A=A$, the following statements hold true:

• (a) Let $B\in \mathbb {C}^{{p\times r}}$. Then $\mathcal {R}\left( {B}\right) \subseteq \mathcal {R}\left( {A}\right)$ if and only if $AA^{\mathord {+}}B=B$.

• (b) Let $C\in \mathbb {C}^{{r\times q}}$. Then $\mathcal {N}\left( {A}\right) \subseteq \mathcal {N}\left( {C}\right)$ if and only if $CA^{\mathord {+}}A=C$.

Lemma A.15

(cf. [6, Lem. A.19]) Let both $L\in \mathbb {C}^{{p\times p}}$ and $R\in \mathbb {C}^{{q\times q}}$ be invertible matrices. Let $M\in \mathbb {C}^{{p\times q}}$ and let $N:=LMR^{-1}$. Then $X:=MR^{-1}$ can be represented via $X=L^{-1}N$ and the Moore–Penrose inverse of X admits the representation $X^{\mathord {+}}=N^{\mathord {+}}NRM^{\mathord {+}}$.

Lemma A.16

([6, Lem. A.26]) Let $E\in \mathbb {C}^{{q\times q}}$, let $b\in \mathbb {R}\backslash \left\{ {0} \right\}$, let $B:=b^{-1}\Im E$, and let $P:=\mathbb {P}_{\mathcal {R}\left( {E}\right) }$ and $Q:=\mathbb {P}_{\mathcal {N}\left( {E}\right) }$. Suppose $B\in \mathbb {C}_\succcurlyeq ^{{q\times q}}$ and $\mathcal {R}\left( {B}\right) =\mathcal {R}\left( {E}\right)$. Let $C\in \mathbb {C}^{{q\times q}}$ and let $S:=E\sqrt{B}^{\mathord {+}}-E^*\sqrt{B}^{\mathord {+}}CP$ and $T:=\sqrt{B}^{\mathord {+}}-\sqrt{B}^{\mathord {+}}CP+Q$. Then

$$\begin{aligned} {{\,\textrm{rank}\,}}\begin{bmatrix}S\\ T\end{bmatrix}&=q,&PS&=S,&SP&=S,&TP&=T-Q \end{aligned}$$

as well as $b^{-1}\Im \left( {T^*S}\right) =P^*\left[ {I_{q}-\left( {PC}\right) ^*\left( {PC}\right) } \right] P$ and $\sqrt{B}^{\mathord {+}}\left( {S-ET}\right) \left( {S-E^*T}\right) ^{\mathord {+}}\sqrt{B}=PCP$. Furthermore, if C is contractive, then $b^{-1}\Im \left( {T^*S}\right)$ is non-negative Hermitian.

Appendix B: Some Facts on the Integration Theory of non-negative Hermitian Measures

Remark B.1

Let $\left( {\Omega ,\mathfrak {A}}\right)$ be a measurable space, let $\mu =\left[ {\mu _{jk}} \right] _{j,k=1}^{q}\in {\mathcal {M}_\succcurlyeq ^{q}(\Omega ,{\mathfrak {A}})}$, and let $f :\Omega \rightarrow \mathbb {C}$ be an ${\mathfrak {A}}$-${\mathfrak {B}_{\mathbb {C}}}$-measurable mapping. For each $U\in \mathbb {C}^{{q\times p}}$, then $U^*\mu U$ belongs to ${\mathcal {M}_\succcurlyeq ^{p}(\Omega ,{\mathfrak {A}})}$. Using standard arguments of measure and integration theory, easily one can see moreover that the following statements are equivalent:

• (i) $f\in \mathcal {L}^{1}\left( {\Omega ,\mathfrak {A},\mu ;\mathbb {C}}\right)$.

• (ii) $f\in \mathcal {L}^{1}\left( {\Omega ,\mathfrak {A},\tau ;\mathbb {C}}\right)$, where $\tau$ is the trace measure of $\mu$.

• (iii) $f\in \mathcal {L}^{1}\left( {\Omega ,\mathfrak {A},U^*\mu U;\mathbb {C}}\right)$ for each $U\in \mathbb {C}^{{q\times p}}$.

If (i) holds true, then $U^*\left( {\int _\Omega f\textrm{d}\mu }\right) U=\int _\Omega f\textrm{d}\left( {U^*\mu U}\right)$ for all $U\in \mathbb {C}^{{q\times p}}$.

Remark B.2

Let $(\Omega ,{\mathfrak {A}})$ be a measurable space, let $\mu \in {\mathcal {M}_\succcurlyeq ^{q}(\Omega ,{\mathfrak {A}})}$, let $N\in {\mathfrak {A}}$ be such that $\mu (N)=O_{{q\times q}}$, and let $f:\Omega \rightarrow \mathbb {C}$ be ${\mathfrak {A}}$-${\mathfrak {B}_{\mathbb {C}}}$-measurable. Then $1_{N}\in \mathcal {L}^{1}\left( {\Omega ,\mathfrak {A},\mu ;\mathbb {C}}\right)$ and $\int _Nf\textrm{d}\mu =O_{{q\times q}}$. Furthermore, $f\in \mathcal {L}^{1}\left( {\Omega ,\mathfrak {A},\mu ;\mathbb {C}}\right)$ if and only if $1_{\Omega \backslash N}f\in \mathcal {L}^{1}\left( {\Omega ,\mathfrak {A},\mu ;\mathbb {C}}\right)$. If $f\in \mathcal {L}^{1}\left( {\Omega ,\mathfrak {A},\mu ;\mathbb {C}}\right)$, then $\int _{\Omega \backslash N}f\textrm{d}\mu =\int _\Omega f\textrm{d}\mu$.

Remark B.3

Let $\left( {\Omega ,{\mathfrak {A}}}\right)$ be a measurable space, let ${\tilde{\Omega }}\in {\mathfrak {A}}\backslash \left\{ {\emptyset } \right\}$, let ${\tilde{{\mathfrak {A}}}}:=\left\{ {A\cap {\tilde{\Omega }}}:{A\in {\mathfrak {A}}}\right\}$, and let ${\tilde{\mu }}\in {\mathcal {M}_\succcurlyeq ^{q}({\tilde{\Omega }},{\tilde{{\mathfrak {A}}}})}$. Then the mapping $\mu :{\mathfrak {A}}\rightarrow \mathbb {C}^{{q\times q}}$ given by $\mu \left( {A}\right) :={\tilde{\mu }}\left( {A\cap {\tilde{\Omega }}}\right)$ belongs to ${\mathcal {M}_\succcurlyeq ^{q}(\Omega ,{\mathfrak {A}})}$ and fulfills ${{\,\textrm{Rstr}\,}}_{{\tilde{{\mathfrak {A}}}}}\mu ={\tilde{\mu }}$ and $\mu \left( {\Omega \backslash {\tilde{\Omega }}}\right) =O_{{q\times q}}$. Moreover, ${{\,\textrm{Rstr}\,}}_{{\tilde{{\mathfrak {A}}}}}\tau ={\tilde{\tau }}$ with $\tau$ and ${\tilde{\tau }}$ being the trace measures of $\mu$ and ${\tilde{\mu }}$, respectively.

Remark B.4

Let $(\Omega ,{\mathfrak {A}})$ be a measurable space, let $\mu \in {\mathcal {M}_\succcurlyeq ^{q}(\Omega ,{\mathfrak {A}})}$, let $\tilde{\Omega }\in {\mathfrak {A}}\backslash \left\{ {\emptyset } \right\}$, and let $\tilde{{\mathfrak {A}}}:=\left\{ {A\cap \tilde{\Omega }}:{A\in {\mathfrak {A}}}\right\}$. Using Remark B.1 and standard arguments of measure theory, one can easily check that $\tilde{\mu }:={{\,\textrm{Rstr}\,}}_{\tilde{{\mathfrak {A}}}}\mu$ belongs to ${\mathcal {M}_\succcurlyeq ^{q}(\tilde{\Omega },\tilde{{\mathfrak {A}}})}$ and that the following statements hold true:

• (a) If $\tau$ is the trace measure of $\tilde{\mu }$, then $\tilde{\tau }:={{\,\textrm{Rstr}\,}}_{\tilde{{\mathfrak {A}}}}\tau$ is the trace measure of $\tilde{\mu }$.

• (b) Let $f:\Omega \rightarrow \mathbb {C}$ be ${\mathfrak {A}}$-${\mathfrak {B}_{\mathbb {C}}}$-measurable and let ${\tilde{A}}\in \tilde{{\mathfrak {A}}}$. Then $1_{{\tilde{A}}}f\in \mathcal {L}^{1}\left( {\Omega ,\mathfrak {A},\mu ;\mathbb {C}}\right)$ if and only if ${{\,\textrm{Rstr}\,}}_{\tilde{\Omega }}(1_{{\tilde{A}}}f)\in \mathcal {L}^{1}\left( {\tilde{\Omega },\tilde{{\mathfrak {A}}},\tilde{\mu };\mathbb {C}}\right)$. In this case, $\int _{{\tilde{A}}}f\textrm{d}\mu =\int _{{\tilde{A}}}f\textrm{d}\tilde{\mu }$.

Proposition B.5

(see, e. g. [9, Prop. B.5]) Let $(\Omega ,{\mathfrak {A}})$ be a measurable space, let $\mu \in {\mathcal {M}_\succcurlyeq ^{q}(\Omega ,{\mathfrak {A}})}$, and let $c\in \mathcal {L}^{1}\left( {\Omega ,{\mathfrak {A}},\mu ;\mathbb {C}}\right)$ be such that $c(\omega )\in [0,\infty )$ for all $\omega \in \Omega$. Then:

• (a) The function $c\odot \mu :{\mathfrak {A}}\rightarrow \mathbb {C}^{{q\times q}}$ defined by $(c\odot \mu )\left( {A}\right) :=\int _Ac\textrm{d}\mu$ for each $A\in {\mathfrak {A}}$ is a well-defined non-negative Hermitian ${q\times q}$ measure belonging to ${\mathcal {M}_\succcurlyeq ^{q}(\Omega ,{\mathfrak {A}})}$.

• (b) Let $g:\Omega \rightarrow \mathbb {C}$ be ${\mathfrak {A}}$-${\mathfrak {B}_{\mathbb {C}}}$-measurable. Then $g\in \mathcal {L}^{1}\left( {\Omega ,{\mathfrak {A}},c\odot \mu ;\mathbb {C}}\right)$ if and only if $g\cdot c\in \mathcal {L}^{1}\left( {\Omega ,{\mathfrak {A}},\mu ;\mathbb {C}}\right)$. If $g\in \mathcal {L}^{1}\left( {\Omega ,{\mathfrak {A}},c\odot \mu ;\mathbb {C}}\right)$, then $\int _Ag\textrm{d}(c\odot \mu )=\int _A\left( {g\cdot c}\right) \textrm{d}\mu$ for each $A\in {\mathfrak {A}}$.