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.
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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
for every choice of \(j\in \left\{ {0,\dotsc ,m-1} \right\} \) as well as
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
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
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
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
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
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
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 \(z\in \mathbb {C}\backslash \left( {\mathbb {R}\cup \mathcal {E}}\right) \).
-
(III’)
-
(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 \(z\in \mathbb {C}\backslash \left( {\mathbb {R}\cup \mathcal {E}}\right) \).
-
(IV’)
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
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
and
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
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
Taking additionally into account (4.6), we can conclude
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
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
Using Remarks A.1 and A.3, thus we obtain
Regarding \(s_{0}=I_{q}\), then
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
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
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
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
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
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,
as well as
For each \(\tau \in \mathbb {N}_0\cup \left\{ {\infty } \right\} \), let
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
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
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
and
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
![figure a](http://media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs11785-022-01283-y/MediaObjects/11785_2022_1283_Figa_HTML.png)
and
![figure b](http://media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs11785-022-01283-y/MediaObjects/11785_2022_1283_Figb_HTML.png)
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
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
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
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
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
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
is contractive, and the identity
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
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
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
and, for all \(n\in \mathbb {N}_0\), moreover,
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 \(z\in \mathbb {C}\backslash \left( {\mathbb {R}\cup \mathcal {D}_{\mathord {\bullet }}}\right) \).
-
(III’)
-
(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 \(z\in \mathbb {C}\backslash \left( {\mathbb {R}\cup \mathcal {D}_{\mathord {\bullet }}}\right) \).
-
(IV’)
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
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
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
Therefore, part (c) is proved. From part (c) we obtain
for all \(z\in \mathbb {C}\), which implies part (d). Using part (c) and Remarks A.1 and A.3, we conclude
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
Using this and Remark A.1, we obtain
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
and, in the case \(\kappa \ge 1\), moreover, that
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
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
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
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
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
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
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
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
and
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
and
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
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
and
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
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
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
and
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
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
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
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
and
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
and
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
![](http://media.springernature.com/lw475/springer-static/image/art%3A10.1007%2Fs11785-022-01283-y/MediaObjects/11785_2022_1283_Equ437_HTML.png)
and
![](http://media.springernature.com/lw445/springer-static/image/art%3A10.1007%2Fs11785-022-01283-y/MediaObjects/11785_2022_1283_Equ438_HTML.png)
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
as well as
and
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
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
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
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
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
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
Lemma 11.8 yields
and
whereas Lemma 11.10 provides
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
Regarding (11.15), we can multiply (11.18) from the left by \( B_{k}^{-1}\) to obtain
Using (11.19) and \(\mathfrak {h}_{2k}\mathfrak {h}_{2k}^{\mathord {+}}\eta \left( {z}\right) =\eta \left( {z}\right) \), we get
Multiplying both sides of (11.20) from the left by \(v^*\left[ {\theta \left( {z}\right) } \right] ^*\) and using (11.17), we obtain
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
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
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,
In view of (11.14), therefore (11.12) is checked. In order to prove (11.13), we now set
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
and
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
![](http://media.springernature.com/lw432/springer-static/image/art%3A10.1007%2Fs11785-022-01283-y/MediaObjects/11785_2022_1283_Equ439_HTML.png)
and
![](http://media.springernature.com/lw432/springer-static/image/art%3A10.1007%2Fs11785-022-01283-y/MediaObjects/11785_2022_1283_Equ440_HTML.png)
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
![](http://media.springernature.com/lw443/springer-static/image/art%3A10.1007%2Fs11785-022-01283-y/MediaObjects/11785_2022_1283_Equ441_HTML.png)
Taking into account Remark 11.7, (9.1), and (5.5), then
![](http://media.springernature.com/lw507/springer-static/image/art%3A10.1007%2Fs11785-022-01283-y/MediaObjects/11785_2022_1283_Equ442_HTML.png)
follows, where
Using (9.1), (5.5), (5.4), (9.3), and (9.5), we can further conclude
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),
and
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
![](http://media.springernature.com/lw534/springer-static/image/art%3A10.1007%2Fs11785-022-01283-y/MediaObjects/11785_2022_1283_Equ443_HTML.png)
and
![](http://media.springernature.com/lw372/springer-static/image/art%3A10.1007%2Fs11785-022-01283-y/MediaObjects/11785_2022_1283_Equ444_HTML.png)
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
![](http://media.springernature.com/lw320/springer-static/image/art%3A10.1007%2Fs11785-022-01283-y/MediaObjects/11785_2022_1283_Equ78_HTML.png)
Let
as well as
From Lemma 11.12 we have (11.26)–(11.29). By virtue of (4.1), (11.28), and (11.32), we get
![](http://media.springernature.com/lw480/springer-static/image/art%3A10.1007%2Fs11785-022-01283-y/MediaObjects/11785_2022_1283_Equ445_HTML.png)
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
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
Taking into account Notation 10.3, it is readily checked that, for each \(k\in \mathbb {N}_0\) fulfilling \(2k\le \kappa \), then
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
Indeed, using Lemma 12.3, (10.4), Notations 10.6 and 5.5, Lemma 11.3, and Remark 12.6, we can infer
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
and
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
and
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
and
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
and
Lemma 12.8 yields (12.3) and (12.4). By virtue of Lemma 12.3, (12.5), and (12.3), we obtain
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
![](http://media.springernature.com/lw354/springer-static/image/art%3A10.1007%2Fs11785-022-01283-y/MediaObjects/11785_2022_1283_Equ446_HTML.png)
and
![](http://media.springernature.com/lw504/springer-static/image/art%3A10.1007%2Fs11785-022-01283-y/MediaObjects/11785_2022_1283_Equ447_HTML.png)
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
as well as
and
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
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
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
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
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
whereas from [6, Lem. 7.13(b)] we obtain
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
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
Lemma 12.11 yields
and
whereas Lemma 12.13 provides
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
Regarding (12.14), we can multiply (12.17) from the left by \( B_{k}^{-1}\) to obtain
Using (12.18) and \(\mathfrak {h}_{{\alpha ,2k}}\mathfrak {h}_{{\alpha ,2k}}^{\mathord {+}}\eta \left( {z}\right) =\eta \left( {z}\right) \), we get
Multiplying both sides of (12.19) from the left by \(v^*\left[ {\theta \left( {z}\right) } \right] ^*\) and using (12.16), we obtain
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
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
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,
In view of (12.13), therefore (12.11) is checked. In order to prove (12.12), we now set
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
and
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
for all \(k\in \mathbb {N}\) with \(2k-1\le \kappa -1\) as well as
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
![](http://media.springernature.com/lw484/springer-static/image/art%3A10.1007%2Fs11785-022-01283-y/MediaObjects/11785_2022_1283_Equ448_HTML.png)
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
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
![](http://media.springernature.com/lw416/springer-static/image/art%3A10.1007%2Fs11785-022-01283-y/MediaObjects/11785_2022_1283_Equ449_HTML.png)
Taking into account Remark 12.10, (9.2), and (12.30), then
![](http://media.springernature.com/lw544/springer-static/image/art%3A10.1007%2Fs11785-022-01283-y/MediaObjects/11785_2022_1283_Equ450_HTML.png)
follows, where
Using (9.2), (12.30), (9.3), (12.29), and (9.6), we can conclude
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
where
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
and
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
and
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,
![](http://media.springernature.com/lw356/springer-static/image/art%3A10.1007%2Fs11785-022-01283-y/MediaObjects/11785_2022_1283_Equ113_HTML.png)
Let
as well as
According to Lemma 12.15, we have (12.25)–(12.28). By virtue of (4.1), (12.27), and (12.35), we obtain
![](http://media.springernature.com/lw514/springer-static/image/art%3A10.1007%2Fs11785-022-01283-y/MediaObjects/11785_2022_1283_Equ451_HTML.png)
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
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
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,
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
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
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
![](http://media.springernature.com/lw385/springer-static/image/art%3A10.1007%2Fs11785-022-01283-y/MediaObjects/11785_2022_1283_Equ452_HTML.png)
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
![](http://media.springernature.com/lw475/springer-static/image/art%3A10.1007%2Fs11785-022-01283-y/MediaObjects/11785_2022_1283_Equ453_HTML.png)
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
![](http://media.springernature.com/lw524/springer-static/image/art%3A10.1007%2Fs11785-022-01283-y/MediaObjects/11785_2022_1283_Equ454_HTML.png)
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
![](http://media.springernature.com/lw469/springer-static/image/art%3A10.1007%2Fs11785-022-01283-y/MediaObjects/11785_2022_1283_Equ455_HTML.png)
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
![](http://media.springernature.com/lw521/springer-static/image/art%3A10.1007%2Fs11785-022-01283-y/MediaObjects/11785_2022_1283_Equ456_HTML.png)
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
According to Notation 10.6, in particular
holds true for all \(n\in \mathbb {N}_0\) fulfilling \(2n\le \kappa \) and all \(z\in \mathbb {C}\) and
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
Proof
Let \(\zeta :=z-\alpha \). Using Lemma 14.2(b), Remarks 12.7 and 12.6, and Lemma 12.3, we obtain
![](http://media.springernature.com/lw487/springer-static/image/art%3A10.1007%2Fs11785-022-01283-y/MediaObjects/11785_2022_1283_Equ121_HTML.png)
Furthermore, (14.3), Lemma 11.3, (11.1), and (12.1) yield
![](http://media.springernature.com/lw456/springer-static/image/art%3A10.1007%2Fs11785-022-01283-y/MediaObjects/11785_2022_1283_Equ122_HTML.png)
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
Proof
Using Lemma 14.2(a), Lemma 11.3, and (11.1), we obtain
Setting \(\zeta :=z-\alpha \) and taking additionally into account (10.4), Notation 10.6, and (14.3), we can conclude then
![](http://media.springernature.com/lw517/springer-static/image/art%3A10.1007%2Fs11785-022-01283-y/MediaObjects/11785_2022_1283_Equ457_HTML.png)
\(\square \)
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
![](http://media.springernature.com/lw315/springer-static/image/art%3A10.1007%2Fs11785-022-01283-y/MediaObjects/11785_2022_1283_Equ458_HTML.png)
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
![](http://media.springernature.com/lw454/springer-static/image/art%3A10.1007%2Fs11785-022-01283-y/MediaObjects/11785_2022_1283_Equ459_HTML.png)
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
For each \(n\in \mathbb {N}_0\) fulfilling \(2n+2\le \kappa \) and all \(z\in \mathbb {C}\), then
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
![](http://media.springernature.com/lw531/springer-static/image/art%3A10.1007%2Fs11785-022-01283-y/MediaObjects/11785_2022_1283_Equ125_HTML.png)
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
![](http://media.springernature.com/lw536/springer-static/image/art%3A10.1007%2Fs11785-022-01283-y/MediaObjects/11785_2022_1283_Equ126_HTML.png)
holds true. Due to (9.4), yielding
equation (14.10) can be further simplified to
![](http://media.springernature.com/lw480/springer-static/image/art%3A10.1007%2Fs11785-022-01283-y/MediaObjects/11785_2022_1283_Equ460_HTML.png)
Using additionally Notation 14.6, (9.3), (9.4), and (14.7) we consequently obtain
![](http://media.springernature.com/lw536/springer-static/image/art%3A10.1007%2Fs11785-022-01283-y/MediaObjects/11785_2022_1283_Equ127_HTML.png)
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
![](http://media.springernature.com/lw441/springer-static/image/art%3A10.1007%2Fs11785-022-01283-y/MediaObjects/11785_2022_1283_Equ128_HTML.png)
follows, where
and
Due to (14.13), (9.4), and (14.7), we have
whereas from (14.14), (9.4), and (14.7), we conclude
From Notation 14.6, (14.12), and (14.15) we get
![](http://media.springernature.com/lw509/springer-static/image/art%3A10.1007%2Fs11785-022-01283-y/MediaObjects/11785_2022_1283_Equ133_HTML.png)
Let
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
Lastly, regarding (14.18), (14.17), and (14.16) and taking into account (9.3), (9.4), and (14.7), we conclude
Hence, in view of (14.18), we have
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
Remark 9.3 shows
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
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
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
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,
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
that
and that
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
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
and
Setting
we see, in view of (14.29), (14.30), and (14.31), that
satisfy \(\det {\mathfrak {R}}\ne 0\) and \(\det {\textbf{R}}\ne 0\) as well as
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
Using additionally (14.33) and \(\det {\mathfrak {R}}\ne 0\), we can thus apply Proposition 6.11 to infer that
is a contractive matrix which fulfills
By virtue of (14.36), (14.22), (14.32), and (14.27), we discern
The application of Lemma 6.9 to the sequence \((s_j)_{j=0}^{\infty }\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,\infty }\) yields
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
Using additionally \(\det {\textbf{R}} \ne 0\), (14.33), (14.19), and Lemma 14.2(a), we then obtain
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
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
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
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,
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
that
and that
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
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
and
Setting
we see, in view of (14.49), (14.50), and (14.51), that
satisfy \(\det {\mathfrak {R}}_{\mathord {\circ }}\ne 0\) and \(\det {\textbf{R}}_{\mathord {\circ }}\ne 0\) as well as
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
Using additionally (14.53) and \(\det {\mathfrak {R}}_{\mathord {\circ }}\ne 0\), we can thus apply Proposition 6.11 to infer that the matrix
fulfills
By virtue of (14.56), (14.42), (14.52), and (14.47), we discern
The application of Lemma 6.9 to the sequence \((a_j)_{j=0}^{\infty }\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,\infty }\) yields
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
Using additionally \(\det {\textbf{R}}_{\mathord {\circ }}\ne 0\), (14.53), (14.19), and Lemma 14.2(b), we then obtain
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
![figure c](http://media.springernature.com/lw685/springer-static/image/art%3A10.1007%2Fs11785-022-01283-y/MediaObjects/11785_2022_1283_Figc_HTML.png)
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
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
In view of (14.28), (14.19), and Remark A.10, we can infer
Lemma 14.4 then shows
Consequently, applying additionally (14.60) and (14.63), we conclude
Part 4: Regarding (14.62) and (14.61), we have
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
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
Consequently, in view of (14.65), we infer
Therefore,
and
In view of \((s_j)_{j=0}^{\infty }\in \mathcal {K}^{\succcurlyeq ,\textrm{e}}_{q,\infty ,\alpha }\), Remark 9.3 yields
If \(n\ge 1\), then from (14.7) and (14.70) we can see that
and, for each \(m\in \left\{ {2n+1,2n+2} \right\} \), moreover,
are fulfilled. Regarding (14.7) also in the case \(n=0\), we can conclude then in general
Using (14.48), (14.19), and Remark A.10, we can infer
Regarding additionally (14.20), the second identity in (14.71), (14.70), and (14.55), then
![](http://media.springernature.com/lw523/springer-static/image/art%3A10.1007%2Fs11785-022-01283-y/MediaObjects/11785_2022_1283_Equ188_HTML.png)
follows. In view of (14.64), we can infer from Remark 9.3 that
From (14.68) we thus can conclude
In view of (14.55) and (14.72), then
follows. Furthermore, (14.69), (14.70), and the first identity in (14.71) yield
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
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
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
Part 5: Let \(\pi ,\rho :\mathbb {C}\rightarrow \mathbb {C}^{{q\times q}}\) be defined by
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
as well as
and
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
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
whereas (14.85), (14.84), and (14.79) provide
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
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
Since \(\mathfrak {k}_{2n+2}=\mathfrak {h}_{2n+2}=L_{n+1}\), we consequently get
and
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
Proof
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
Proof
In view of Lemma 14.2(a) as well as Lemma 11.3 and (11.1), we have
![](http://media.springernature.com/lw512/springer-static/image/art%3A10.1007%2Fs11785-022-01283-y/MediaObjects/11785_2022_1283_Equ461_HTML.png)
Using (10.4) and Notation 14.3, then we get finally
\(\square \)
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
Proof
Let \(\zeta :=z-\alpha \). Using Lemma 14.2(b) as well as Remarks 12.7 and 12.6, we obtain
![](http://media.springernature.com/lw491/springer-static/image/art%3A10.1007%2Fs11785-022-01283-y/MediaObjects/11785_2022_1283_Equ462_HTML.png)
In view of (14.4), Lemmata 12.3 and 11.3, (12.1), and (11.1), we get furthermore
![](http://media.springernature.com/lw400/springer-static/image/art%3A10.1007%2Fs11785-022-01283-y/MediaObjects/11785_2022_1283_Equ463_HTML.png)
Hence,
![](http://media.springernature.com/lw486/springer-static/image/art%3A10.1007%2Fs11785-022-01283-y/MediaObjects/11785_2022_1283_Equ464_HTML.png)
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
For each \(n\in \mathbb {N}_0\) fulfilling \(2n+1\le \kappa \) and all \(z\in \mathbb {C}\), then
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
![](http://media.springernature.com/lw471/springer-static/image/art%3A10.1007%2Fs11785-022-01283-y/MediaObjects/11785_2022_1283_Equ204_HTML.png)
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
![](http://media.springernature.com/lw531/springer-static/image/art%3A10.1007%2Fs11785-022-01283-y/MediaObjects/11785_2022_1283_Equ205_HTML.png)
holds true. Due to (9.4), yielding
equation (15.4) can be further simplified to
![](http://media.springernature.com/lw480/springer-static/image/art%3A10.1007%2Fs11785-022-01283-y/MediaObjects/11785_2022_1283_Equ465_HTML.png)
Using additionally Notation 14.6, (9.3), (9.4), and (15.1) we consequently obtain
![](http://media.springernature.com/lw573/springer-static/image/art%3A10.1007%2Fs11785-022-01283-y/MediaObjects/11785_2022_1283_Equ206_HTML.png)
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
![](http://media.springernature.com/lw384/springer-static/image/art%3A10.1007%2Fs11785-022-01283-y/MediaObjects/11785_2022_1283_Equ207_HTML.png)
follows, where
and
Due to (15.7), (9.4), and (15.1), we have
whereas from (15.8), (9.4), and (15.1), we conclude
From Notation 14.6, (15.6), and (15.9) we get
![](http://media.springernature.com/lw509/springer-static/image/art%3A10.1007%2Fs11785-022-01283-y/MediaObjects/11785_2022_1283_Equ212_HTML.png)
Let
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
Lastly, regarding (15.12), (15.11), and (15.10) and taking into account (9.3), (9.4), and (15.1), we conclude
Hence, in view of (15.12), we have
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
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
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
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,
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
that
and that
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
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
and
Setting
we see, in view of (15.21), (15.22), and (15.23), that
satisfy \(\det {\mathfrak {R}}_{\mathord {\circ }}\ne 0\) and \(\det {\textbf{R}}_{\mathord {\circ }}\ne 0\) as well as
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
In view of (15.25) and \(\det {\mathfrak {R}}_{\mathord {\circ }}\ne 0\), Proposition 6.11 shows then that the matrix
is contractive and fulfills
By virtue of (15.28), (15.14), (15.24), and (15.19), we discern
The application of Lemma 6.9 to the sequence \((a_j)_{j=0}^{\infty }\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,\infty }\) yields
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
Using additionally \(\det {\textbf{R}}_{\mathord {\circ }}\ne 0\), (15.25), (14.19), and Lemma 14.2(b), we then obtain
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
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
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
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,
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
that
and that
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
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
and
Setting
we see, in view of (15.41), (15.42), and (15.43), that
satisfy \(\det {\mathfrak {R}}\ne 0\) and \(\det {\textbf{R}}\ne 0\) as well as
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
Using additionally (15.45) and \(\det {\mathfrak {R}}\ne 0\), the application of Proposition 6.11 yields that the matrix
is contractive and fulfills
By virtue of (15.48), (15.34), (15.44), and (15.39), we discern
The application of Lemma 6.9 to the sequence \((s_j)_{j=0}^{\infty }\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,\infty }\) yields
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
Using additionally \(\det {\textbf{R}} \ne 0\), (15.45), (14.19), and Lemma 14.2(a), we then obtain
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
![](http://media.springernature.com/lw212/springer-static/image/art%3A10.1007%2Fs11785-022-01283-y/MediaObjects/11785_2022_1283_Equ254_HTML.png)
From (15.24) and (15.17), we see \({{\,\textrm{rank}\,}}\bigl [{\begin{matrix}S_{\mathord {\circ }}\\ T_{\mathord {\circ }}\end{matrix}}\bigr ]=q\). Let
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
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
Since the matrix \(\bigl [{\begin{matrix}\eta I_{q}&{}s_{0}\\ O_{{q\times q}}&{} I_{q}\end{matrix}}\bigr ]\) is obviously invertible, then
follows. Using (15.20), (14.19), and Remark A.10, we can infer
Lemma 15.1 then shows
Hence, using additionally (15.52) and (15.56), we conclude
Part 4: Regarding (15.54) and (15.53), we have
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
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
Consequently, in view of (15.58), we infer
Therefore,
and
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
and, for each \(m\in \left\{ {2n,2n+1} \right\} \), moreover,
are fulfilled. Regarding (15.1) also in the case \(n=0\), we can conclude then in general
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
follows. In view of (15.57), we can infer from (14.70) that
From (15.61) we thus can conclude
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
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
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
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
Part 5: Let \(\pi _{\mathord {\circ }},\rho _{\mathord {\circ }}:\mathbb {C}\rightarrow \mathbb {C}^{{q\times q}}\) be defined by
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
as well as
and
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
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
whereas (15.76), (15.75), and (15.70) provide
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
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
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
Proof
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
For all \(z\in \mathbb {C}\), let
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
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
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
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
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
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,
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
that
and
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
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
and
hold true for all \(z\in \Pi _{\mathord {+}}\). Setting
we see, in view of (16.13) and (16.14), that
satisfy \(\det {\mathfrak {R}}\ne 0\) and \(\det {\textbf{R}}\ne 0\) as well as
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
Using additionally (16.16) and \(\det {\mathfrak {R}}\ne 0\), we can thus apply Proposition 6.11 to infer that
is a contractive matrix which fulfills
By virtue of (16.19), (16.6), (16.15), and (16.11), we discern
The application of Lemma 6.9 to the sequence \((s_j)_{j=0}^{\infty }\in \mathcal {H}^{\succcurlyeq ,\textrm{e}}_{q,\infty }\) yields
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
Using additionally \(\det {\textbf{R}} \ne 0\), (16.16), (16.4), and Lemma 14.2(a), we then obtain
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
Regarding \(\mathfrak {k}_{0}=s_{0}\), we obtain furthermore
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
![](http://media.springernature.com/lw189/springer-static/image/art%3A10.1007%2Fs11785-022-01283-y/MediaObjects/11785_2022_1283_Equ304_HTML.png)
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
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
In view of (16.12), (16.4), and Remark A.10, we can infer
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
Part 4: Regarding (16.27) and (16.26), we have
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
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
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
Part 5: Let \(\pi ,\rho :\mathbb {C}\rightarrow \mathbb {C}^{{q\times q}}\) be defined by
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
as well as
and
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
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
whereas (16.40), (16.39), and (16.34) provide
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
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
Since \(\mathfrak {k}_{0}=\mathfrak {h}_{0}=L_{0}\), we consequently get
and
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
Proof
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The article is based primarily on the dissertation of Michaela Wall. Conceptualization: Bernd Fritzsche; Investigation: Michaela Wall, Bernd Fritzsche, Bernd Kirstein, Conrad Mädler; Project administration: Bernd Fritzsche, Conrad Mädler; Resources: Bernd Fritzsche, Bernd Kirstein; Supervision: Bernd Fritzsche; Validation: Bernd Fritzsche; Writing – original draft: Michaela Wall; Writing – review & editing: Bernd Fritzsche, Bernd Kirstein, Conrad Mädler.
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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
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 \(A\in {\mathfrak {A}}\).
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Wall, M., Fritzsche, B., Kirstein, B. et al. Weyl Sets in a Truncated Matricial Stieltjes Moment Problem. Complex Anal. Oper. Theory 16, 112 (2022). https://doi.org/10.1007/s11785-022-01283-y
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DOI: https://doi.org/10.1007/s11785-022-01283-y