Conrad Mädler

On a Special Parametrization of Matricial α-Stieltjes One-sided Non-negative Definite Sequences

The characterization of the solvability of matrix versions of truncated Stieltjes-type moment problems led to the class of one-sided α-Stieltjes non-negative definite sequences of complex q × q matrices. The study of this class and some of its important subclasses is the central theme of this paper. We introduce an inner parametrization for sequences of complex matrices, which is particularly well-suited to the class of sequences under consideration. Furthermore, several interrelations between this parametrization and the canonical Hankel parametrization are indicated.

On a Simultaneous Approach to the Even and Odd Truncated Matricial Hamburger Moment Problems

The main goal of this paper is to achieve a simultaneous treatment of the even and odd truncated matricial Hamburger moment problems in the most general case. In the odd case, these results are completely new for the matrix case, whereas the scalar version was recently treated by V.A. Derkach, S. Hassi and H.S.V. de Snoo [12]. The even case was studied earlier by G.-N. Chen and Y.-J. Hu [9]. Our approach is based on Schur analysis methods. More precisely, we use two interrelated versions of Schurtype algorithms, namely an algebraic one and a function-theoretic one. The algebraic version was worked out in a former paper of the authors. It is an algorithm which is applied to finite or infinite sequences of complex matrices. The construction and investigation of the function-theoretic version of our Schur-type algorithm is a central theme of this paper. This algorithm will be applied to relevant subclasses of holomorphic matrix-valued functions of the Herglotz–Nevanlinna class. Using recent results on the holomorphicity of the Moore–Penrose inverse of matrix-valued Herglotz–Nevanlinna functions, we obtain a complete description of the solution set of the moment problem under consideration in the most general situation.

On Matrix-valued Stieltjes Functions with an Emphasis on Particular Subclasses

The paper deals with particular classes of q×q matrix-valued functions which are holomorphic in \(\mathbb{C}\backslash [\alpha, +\infty)\), where α is an arbitrary real number. These classes are generalizations of classes of holomorphic complex-valued functions studied by Kats and Krein [17] and by Krein and Nudelman [19]. The functions are closely related to truncated matricial Stieltjes problems on the interval [α+∞). Characterizations of these classes via integral representations are presented. Particular emphasis is placed on the discussion of the Moore–Penrose inverse of these matrix-valued functions.

On a simultaneous approach to the even and odd truncated matricial Stieltjes moment problem II: An α-Schur–Stieltjes-type algorithm for sequences of holomorphic matrix-valued functions

The main goal of this paper is to achieve a simultaneous treatment of the even and odd truncated matricial Stieltjes moment problems in the most general case. These results are generalizations of results of Chen and Hu [5,17] which considered the particular case

On the Structure of Hausdorff Moment Sequences of Complex Matrices

\alpha=0

On Resolvent Matrix, Dyukarev–Stieltjes Parameters and Orthogonal Matrix Polynomials via \([0, \infty )\)-Stieltjes Transformed Sequences

. Our approach is based on Schur analysis methods. More precisely, we use two interrelated versions of Schur-type algorithms, namely an algebraic one and a function-theoretic one. The algebraic version was worked out in a former paper of the authors. It is an algorithm which is applied to finite or infinite sequences of complex matrices. The construction and investigation of the function-theoretic version of our Schur-type algorithm is a central theme of this paper. This algorithm will be applied to relevant subclasses of holomorphic matrix-valued functions of the Stieltjes class. Using recent results on the holomorphicity of the Moore-Penrose inverse of matrix-valued Stieltjes functions, we obtain a complete description of the solution set of the moment problem under consideration in the most general situation.

On the Concept of Invertibility for Sequences of Complex p \times q -matrices and its Application to Holomorphic p \times q -matrix-valued Functions

The paper treats several aspects of the truncated matricial [α, β]-Hausdorff type moment problems. It is shown that each [α, β]-Hausdorff moment sequence has a particular intrinsic structure. More precisely, each element of this sequence varies within a closed bounded matricial interval. The case that the corresponding moment coincides with one of the endpoints of the interval plays a particular important role. This leads to distinguished molecular solutions of the truncated matricial [α, β]-Hausdorff moment problem, which satisfy some extremality properties. The proofs are mainly of algebraic character. The use of the parallel sum of matrices is an essential tool in the proofs.

Rational q × q Carathéodory Functions and Central Non-negative Hermitian Measures

By using Schur transformed sequences and Dyukarev–Stieltjes parameters we obtain a new representation of the resolvent matrix corresponding to the truncated matricial Stieltjes moment problem. Explicit relations between orthogonal matrix polynomials and matrix polynomials of the second kind constructed from consecutive Schur transformed sequences are obtained. Additionally, a non-negative Hermitian measure for which the matrix polynomials of the second kind are the orthogonal matrix polynomials is found.

An Addendum to a Paper by Li and Zhang

The main topic of this paper is the invertibility of finite and infinite sequences of complex\( p \times q \)-matrices.This concept was previously considered in the mathematical literature for the special case in which p=q, under certain regularity conditions, in the context of matricial power series inversion. The problem of describing all (finite and infinite) invertible sequences of complex\( p \times q \)-matrices leads directly to the class of “first term dominant” sequences \( (s_j)^{k}_{j=0} \)of complex \( p \times q \)-matrices.These sequences have the property that the null space of s0 is contained in the null spaces of all sjwhile the range of s0 encompasses the range of every sj The inverse sequence \( (s^{\ddag}_j)^{k}_{j=0} \) in \(\mathbb{C}^{q\times p} \) for an invertible sequence \( (s_j)^{k}_{j=0} \)in \(\mathbb{C}^{q\times p} \) is then constructed. This leads, in conjunction with the concept of power series inversion, to a generalrecursive method for constructing a reciprocal sequence \( (s^{\sharp}_j)^{k}_{j=0} \) in \(\mathbb{C}^{q\times p} \) to any given sequence \( (s_j)^{k}_{j=0} \) in \(\mathbb{C}^{q\times p} \) It is shown that if \( (s_j)^{k}_{j=0} \) is invertible, then its inverse and reciprocal sequences coincide, i.e.,\( (s^{\ddag}_j)^{k}_{j=0} \) = \( (s^{\sharp}_j)^{k}_{j=0} \).Using reciprocal sequences allows for interesting new approaches to a number of fascinating problems in matricial complex analysis. This paper considers the holomorphicity of the Moore-Penrose inverse of a \( p \times q \) -matrixfunction, using an approach based on analyzing the structure of Taylor coefficient sequences. A main result of this paper states that the Moore-Penrose inverse \( F^{\dag} \) of a complex \( p \times q \)-matrix-function F which is holomorphic in an open disk K of the complex plane \( \mathbb{C} \) is holomorphic in K if, and only if, the Taylor-McLaurin coefficient sequence \( (s_j)^{k}_{j=0} \) for F at the center \( {Z_0} \) of K is invertible. When this is the case, the reciprocal sequence \( (s^{\ddag}_j)^{k}_{j=0} \) to \( (s_j)^{k}_{j=0} \) is the Taylor-McLaurin coefficient sequence for \( F^{\dag} \) in \( {Z_0} \)

Matricial Canonical Moments and Parametrization of Matricial Hausdorff Moment Sequences

We give an explicit representation of central measures corresponding to finite Toeplitz non-negative definite sequences of complex q × q matrices. Such measures are intimately connected to central q × q Caratheodory functions. This enables us to prove an explicit representation of the nonstochastic spectral measure of an arbitrary multivariate autoregressive stationary sequence in terms of the covariance sequence.