Parametrization of the Solution Set of a Matricial Truncated Hamburger Moment Problem by a Schur Type Algorithm

Abstract

This paper contains a Schur analytic approach to a truncated matricial moment problem of Hamburger type, which is studied in the most general case. It is shown that a Schur type algorithm constructed by the authors for a related moment problem can be suitably modified to obtain a full description of the solution set with the aid of a linear fractional transformation with polynomial generating matrix-valued function. The main feature of our Schur type algorithm consists of an appropriate synthesis of two different versions of types of algorithms, namely on the one side an algebraic one working for sequences of complex matrices and on the other side a function theoretic one applied to special classes of holomorphic matrix functions.

Appendices

Appendix A Some Particular Facts on Matrix Theory

Remark A.1

Let and let be such that . Then and .

For each , there exists a unique matrix X such that the four equations AXA = A, XAX = X, , and (XA)^∗ = XA hold true. This particular matrix X is said to be the Moore–Penrose inverse of A and one writes for this matrix X. In particular, if A is a non-singular complex q × q matrix, then . It seems to be useful stating some basic results on Moore–Penrose inverses of complex matrices.

Remark A.2

If , then .

Lemma A.3 (see e. g. [8, Lem. 1.1.9])

Let and let

$$\displaystyle \begin{aligned} E =\begin{pmatrix} a & b \\ c & d \end{pmatrix} \end{aligned} $$

(A.1)

be the block representation of E with p × p block a. Then the matrix E is non-negative Hermitian if and only if the four conditions, , c = b ^∗ , andare fulfilled.

A complex p × q matrix K is called contractive in case ∥K∥S ≤ 1.

Remark A.4

Let . Using Lemma A.3 (see also, e. g., [8, Thm. 1.1.2]), easily one can see that the matrix K is contractive if and only if the matrix I _q − K ^∗ K is non-negative Hermitian.

Remark A.5

Let $\widetilde {J}_{q}$ be given by (4.1). Let be such that

(A.2)

In view of

then . In particular, if and only if .

Remark A.6

Let be such that (A.2) and hold true. In view of Remark A.5, then . Because of and Remark A.4, moreover, the matrix is contractive.

Remark A.7

Let C be a contractive complex q × q matrix. Let and . Because of

then . Regarding Remark 4.1 and that C is contractive, furthermore

Clearly, .

We will write for the (left) Euclidean inner product in , i. e., for all , let . If is a non-empty subset of , then the set of all which fulfil for all is a subspace of and is called (left) orthogonal complement of . If and are subspaces of such that for every choice of u in and w in , then and is said to be the orthogonal sum of and . If is a subspace of , then there exists exactly one matrix such that both and are fulfilled for each . This matrix is called the orthoprojection matrix onto . In particular, for all . A complex q × q matrix P is said to be an orthogonal projection matrix, if there exists a subspace of such that .

Proposition A.8

Let . Then P is an orthogonal projection matrix if and only if P ² = P and P ^∗ = P hold true.

For a detailed proof of Proposition A.8, see, e. g., [34, Satz 2.54].

Remark A.9

If is a subspace of , then .

Lemma A.10

Let and let be a sequence of linear subspaces of such that

(A.3)

holds true. For all , let and let . Then

$$\displaystyle \begin{aligned} \left[\sum_{l=1}^m\eta_l(P_l-P_{l-1}) \right]\left[\sum_{l=1}^m\frac{1}{\eta_l}(P_l-P_{l-1}) \right] =I_{q}. \end{aligned} $$

Proof

From (A.3) and a well-known result on orthoprojection matrices (see, e. g., [34, Satz 4.31]) we get that holds true for all . Moreover, (A.3) and a further well-known result on orthoprojection matrices (see, e. g., [34, Satz 4.30(c)]) deliver the equations for all and the representation as orthogonal sum. Therefore, using the Kronecker delta δ _jk, it is readily checked that (P _j − P _j−1)(P _k − P _k−1) = δ _jk(P _j − P _j−1) for all and, consequently,

$$\displaystyle \begin{aligned} &\left[\sum_{l=1}^m\eta_l(P_l-P_{l-1}) \right]\left[\sum_{l=1}^m\frac{1}{\eta_l}(P_l-P_{l-1}) \right]\\ &=\sum_{j=1}^m\sum_{k=1}^m\eta_j\frac{1}{\eta_k}(P_j-P_{j-1})(P_k-P_{k-1}) =\sum_{j=1}^m\sum_{k=1}^m\eta_j\frac{1}{\eta_k}\delta_{jk}(P_j-P_{j-1})\\ &=\sum_{l=1}^m\eta_l\frac{1}{\eta_l}(P_l-P_{l-1}) =\sum_{l=1}^m(P_l-P_{l-1}) =P_m-P_0 =I_{q}-0_{{q\times q}} =I_{q}. \end{aligned}$$

□

Remark A.11

Let be a subspace of with dimension . Let u ₁, u ₂, …, u _d be an orthonormal basis of and let . Then .

Remark A.12

For each , the equations as well as hold true.

Remark A.13

If , then and .

Lemma A.14

Let . Then:

• (a) Let . Thenif and only if.

• (b) Let . Thenif and only if.

Remark A.15

Let be such that fulfils 1 ≤ r ≤ q − 1. Let u ₁, u ₂, …, u _q be an orthonormal basis of such that u ₁, u ₂, …, u _r is a basis of , let , and let . For every choice of , then and .

Lemma A.16

Let , letbe such that U ^∗ U = I _r , and let . Then the matricesandfulfilandas well as$B^* A=U\widetilde {B}^*\widetilde {A}U^*$ . In particular,if and only if.

Proof

Clearly, . Moreover, we have $A^* A=U\widetilde {A}^* U^* U\widetilde {A}U^*=U\widetilde {A}^*\widetilde {A}U^*$. Obviously, . Taking additionally into account Proposition A.8, we consequently obtain

(A.4)

as well as

It remains to show that is fulfilled. In view of (A.4), we have

(A.5)

If r = q, then U is unitary and, therefore, and the assertion follows from (A.5). Let r < q and . Then there is such that is a unitary q × q matrix. In particular,

$$\displaystyle \begin{aligned} \begin{pmatrix} U^* U&U^* V\\ V^* U&V^* V \end{pmatrix} =W^* W =\begin{pmatrix} I_{r} & 0_{{r\times d}}\\ 0_{{d\times r}} & I_{d} \end{pmatrix} \text{ and } UU^*+VV^*=WW^*=I_{q} \end{aligned} $$

(A.6)

hold true. Using U ^∗ U = I _r and Remarks A.11 and A.9, then and follow. Additionally using (A.5) and (A.6), we obtain

□

Lemma A.17

Letbe such thatfulfils r ≥ 1. Let u ₁, u ₂, …, u _rbe an orthonormal basis ofand let . Furthermore, let P and Q be complex q × q matrices such thatas well asandhold true. Then the matrixis non-singular and the matricesandfulfil and ψ ^∗ ϕ = (B ⁻¹ U)^∗(Q ^∗ P)(B ⁻¹ U). Furthermore, the matricesandfulfil the following statements:

• (a) and$\det (S^* S+T^* T)=\det (\phi ^*\phi +\psi ^*\psi )$.

• (b) T ^∗ S = B ^−∗(Q ^∗ P)B ⁻¹.

• (c) S = PB ⁻¹and T = QB ⁻¹as well as.

Proof

The idea of the proof is taken from [2, Lem. 4.3]. We only consider the case r < q. Let . Then there is such that is a unitary q × q matrix. In particular, (A.6) holds true. Using Remark 4.1, we get . Set , then Remark A.6 shows that $\det B\neq 0$ and that is contractive. Since W is unitary, then the matrix is contractive as well. Moreover, we have

Obviously, and B + A = 2Q are true and, consequently,

(A.7)

follow. According to Remark A.11, we have . Thus, yields UU ^∗ P = P. Therefore, (A.7) and (A.6) imply , i. e., V ^∗ C = V ^∗. Considering (A.6), then the lower blocks of K read V ^∗ CU = V ^∗ U = 0_d×r and V ^∗ CV = V ^∗ V = I _d which is, in particular, unitary. Consequently, K admits the block representation

$$\displaystyle \begin{aligned} K = \begin{pmatrix} U^* CU & 0_{{r\times d}}\\ 0_{{d\times r}} & I_{d} \end{pmatrix}. \end{aligned} $$

(A.8)

Using (A.6) and (A.8), we get

and, therefore,

Due to (A.7), we infer

$$\displaystyle \begin{aligned} PB^{-1}&=UU^* PB^{-1} UU^*=U\phi U^* \end{aligned} $$

and

$$\displaystyle \begin{aligned} QB^{-1}&=UU^* QB^{-1} UU^*+VV^*=U\psi U^*+VV^*. \end{aligned} $$

In view of (A.6) and and using Remark A.9, we have . Consequently, PB ⁻¹ = S and QB ⁻¹ = T hold true which proves (c) and . Assertion (b) immediately follows from (c). Using U ^∗ U = I _r and , the application of Lemma A.16 yields $\det (S^* S+T^* T)=\det (\phi ^*\phi +\psi ^*\psi )$ and . Moreover, considering that T ^∗ S = U(ψ ^∗ ϕ)U ^∗ holds true, finally we obtain

$$\displaystyle \begin{aligned} \psi^*\phi =U^* U(\psi^*\phi)U^* U &=U^* T^* SU\\ &=U^*(QB^{-1} )^*(PB^{-1} )U =(B^{-1} U)^*(Q^* P)(B^{-1} U) \end{aligned}$$

which completes the proof. □

Appendix B Some Facts on the Integration Theory of Non-negative Hermitian Measures

In this section, we present basic facts regarding the integration theory with respect to non-negative Hermitian measures. Throughout the section, let . We write denoting the σ-algebra of all Borel subsets of . Let Ω be a non-empty set and let be a σ-algebra on Ω. Consider a measure ν on the measurable space . We use to denote the set of all --measurable functions such that . We will write for the σ-algebra of all Borel subsets of . An --measurable function is said to be integrable with respect to ν if belongs to , i. e. all entries f _jk belong to the class . In this case, let

A matrix-valued function μ the domain of which is and the values of which belong to the set of all non-negative Hermitian complex q × q matrices is called non-negative Hermitian q × q measure on if it is σ-additive, i. e., if μ fulfils for each sequence of pairwise disjoint sets belonging to . By we denote the set of all non-negative Hermitian q × q measures on , i. e., the set of all σ-additive mappings . Let . For each and for each , the function μ _jk describes a complex measure on and the variation ν _jk of μ _jk is a finite measure on . Especially, μ ₁₁, μ ₂₂, …, μ _qq and the so-called trace measure of μ are finite measures on . For each function f belonging to we use the notation

For this integral, we write as well.

Lemma B.1

Let be a measurable space, let , and let be an - -measurable mapping. Using standard arguments of measure and integration theory, easily one can see that the following statements are equivalent:

• (i) .

• (ii) .

• (iii) , where τ is the trace measure of μ.

• (iv) for each.

Now we turn our attention to an other integral based on investigations by I. S. Kats [26] and M. Rosenberg [31]. Let be a measurable space and let . Then, for every choice of j and k in , the complex measure μ _jk is absolutely continuous with respect to the trace measure τ of μ. If ν describes an arbitrary measure on such that, for all , the complex measure μ _jk is absolutely continuous with respect to ν, we say that μ is absolutely continuous with respect to ν and the matrix-valued function built by the corresponding Radon–Nikodym derivatives of μ _jk with respect to ν is said to be a version of the Radon–Nikodym derivative of μ with respect to ν and is well defined up to sets of zero ν-measure. An ordered pair consisting of an --measurable function and an --measurable function is said to be left-integrable with respect to μ if $\Phi \mu _{\tau }^{\prime }\Psi ^*$ belongs to . In this case the integral

is (well) defined and we also write for this integral.

Appendix C Linear Fractional Transformations of Matrices

In this appendix we summarize some basic facts on linear fractional transformations of matrices. Our considerations modify results due to V. P. Potapov [30], stated in [20].

Remark C.1

Let and . Then easily one can check that the following statements are equivalent (see, e. g., [20, Lem. D.2]):

• (i) The set is non-empty.

• (ii) The set is non-empty.

• (iii) .

Notation C.2

Let and let (A.1) be the block representation of E with p × p block a. If , then the linear fractional transformations and are defined by

Proposition C.3

Let and let

be the block representations of E ₁and E ₂with p × p blocks a ₁and a ₂ . Letand let (A.1) be the block representation of E with p × p block a. Suppose thatandhold true. Let . Then . Furthermore, if , thenfor all.

A detailed proof of Proposition C.3 is given, e. g., in [20, Prop. D.4]. Note that the conditions and do not imply (see [20, Example D.6]).