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Conrad Mädler

Publications and source records attributed to Conrad Mädler.

At least 19 recordsLinked to original sources
arXiv2023

Characteristic Function, Schur Interpolation Problem and Darlington Synthesis

In this paper we would like to show the interrelation between the different mathematical theories concerning the Schur interpolation problem, contractions in Hilbert spaces, pseudocontinuation and Darlington synthesis. The main objects of this article are contractive functions holomorphic in the unit disc (Schur functions). Here they are considered, on the one hand, as characteristic functions of contractions in Hilbert spaces and, on the other hand, as transfer functions of open systems.

math.CV↗
arXiv2023

Characteristic Function, Schur Parameters and Pseudocontinuation of Schur functions

In [19] there is an approach to the investigation of the pseudocontinuability of Schur functions in terms of Schur parameters. In particular, there was obtained a criterion for the pseudocontinuability of Schur functions and the Schur parameters of rational Schur functions were described. This approach is based on the description in terms of the Schur parameters of the relative position of the largest shift and the largest coshift in a completely nonunitary contraction. It should be mentioned that these results received a further development in [8, 21-24]. This paper is aimed to give a survey about essential results on this direction. The main object in the approach is based on considering a Schur function as characteristic function of a contraction (see Section 1.2). This enables us outgoing from Schur parameters to construct a model of the corresponding contraction (see Section 2). In this model, the relative position of the largest shift and the largest coshift in a completely nonunitary contraction is described in Section 3 and then, based on this model, to find characteristics which are responsible for the pseudocontinuability of Schur functions (see Sections 4 and 5). The further parts of this paper (see Sections 6-8) admit applications of the above results to the study of properties of Schur functions and questions related with them.

math.CV↗
arXiv2023

Characteristic function of M. S. Livšic and triangular models of bounded linear operators

This paper is dedicated to the introduction in a circle of ideas and methods, which are connected with the notion of characteristic function of a non-selfadjoint operator. We start with the consideration of closed and open systems (Subsections 2.1.1-2.1.2). In Subsections 2.1.2-2.1.3 we introduce the notion of operator colligation and define the characteristic function of the operator colligation as transfer function of the corresponding open system. In Section 3 we state three basic properties of the c.o.f.. First (Subsection 3.1), we note that the c.o.f. is the full unitary invariant of the operator colligation. Second (see Theorem 3.4), it turns out that the invariant subspaces of the corresponding operator are associated with left divisors of the c.o.f.. Third, the $J$-property of the c.o.f. (see (3.6)-(3.8)) is a basic property which determines the class of c. o. f. (see Section 4). In Chapter 4 we describe the classes of characteristic functions which play an important role in our considerations. In Chapter 5 we state necessary facts on multiplicative integral. Chapter 6 is devoted to the factorization theorem (Theorem 6.7) for matrix-valued characteristic function. In Chapter 7 we construct a triangular Livšic model of bounded linear operator and as application we obtain some known results on dissipative operators.

math.CV↗
arXiv2023

On the Matricial Truncated Moment Problem. II

We continue the study of truncated matrix-valued moment problems begun in arXiv:2310.00957. Let $q\in\mathbb{N}$. Suppose that $(\mathcal{X},\mathfrak{X})$ is a measurable space and $\mathcal{E}$ is a finite-dimensional vector space of measurable mappings of $\mathscr{X}$ into $\mathcal{H}_q$, the Hermitian $q\times q$ matrices. A linear functional $Λ$ on $\mathcal{E}$ is called a moment functional if there exists a positive $\mathcal{H}_q$-valued measure $μ$ on $(\mathcal{X},\mathfrak{X})$ such that $Λ(F)=\int_\mathcal{X} \langle F, \mathrm{d}μ\rangle$ for $F\in \mathcal{E}$. In this paper a number of special topics on the truncated matricial moment problem are treated. We restate a result from (Mourrain and Schmüdgen, 2016) to obtain a matricial version of the flat extension theorem. Assuming that $\mathcal{X}$ is a compact space and all elements of $ \mathcal{E}$ are continuous on $\mathcal{X}$ we characterize moment functionals in terms of positivity and obtain an ordered maximal mass representing measure for each moment functional. The set of masses of representing measures at a fixed point and some related sets are studied. The class of commutative matrix moment functionals is investigated. We generalize the apolar scalar product for homogeneous polynomials to the matrix case and apply this to the matricial truncated moment problem.

math.FA↗
arXiv2023

On the Truncated Matricial Moment Problem. I

This paper is about the general truncated matrix-valued moment problem. Let $\mathcal{H}_q$ denote the complex Hermitian $q\times q$-matrices, $q\in \mathbb{N}$. Suppose that $(\mathcal{X},\mathfrak{X})$ is a measurable space and $\mathcal{E}$ is a finite-dimensional vector space of measurable mappings of $\mathcal{X}$ into $\mathcal{H}_q$. A linear functional $Λ$ on $\mathcal{E}$ is called a moment functional if there exists a positive $\mathcal{H}_q$-valued measure $μ$ on $(\mathcal{X},\mathfrak{X})$ such that $Λ(F)=\int_\mathcal{X} \langle F,\mathrm{d}μ\rangle$ for $F\in \mathcal{E}$. We prove a matricial version of the Richter-Tchakaloff theorem which states that each moment functional on $\mathcal{E}$ has a finitely atomic representing measure. It is shown that strictly positive linear functionals on $\mathcal{E}$ are moment functionals. For a moment functional $Λ$, we study the set of atoms $\mathcal{W}(Λ)$ and the Carathéodory numbers $\mathrm{Car}(Λ)$, $\mathrm{car}(Λ)$ and we define and investigate the core set $\mathcal{V}(Λ)$. A main result of the paper is the equality $\mathcal{W}(Λ)=\mathcal{V}(Λ)$.

math.FA↗
arXiv2021

The Weyl matrix balls corresponding to the matricial truncated Hamburger moment problem

The main goal of the paper is to parametrize the Weyl matrix balls associated with an arbitrary matricial truncated Hamburger moment problem. For the special case of a non-degenerate matricial truncated Hamburger moment problem the corresponding Weyl matrix balls were computed by I. V. Kovalishina in the framework of V. P. Potapov's method of "Fundamental matrix inequalities".

math.CA↗
arXiv2020

A Schur-Nevanlinna type algorithm for the truncated matricial Hausdorff moment problem

The main goal of this paper is to achieve a parametrization of the solution set of the truncated matricial Hausdorff moment problem in the non-degenerate and degenerate situation. We treat the even and the odd cases simultaneously. 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, worked out in our former paper arXiv:1908.05115, is an algorithm which is applied to finite or infinite sequences of complex matrices. The construction and discussion of the function-theoretic version is a central theme of this paper. This leads us to a complete description via Stieltjes transform of the solution set of the moment problem under consideration. Furthermore, we discuss special solutions in detail.

math.CA↗
arXiv2019

Schur analysis of matricial Hausdorff moment sequences

We develop the algebraic instance of an algorithmic approach to the matricial Hausdorff moment problem on a compact interval $[α,β]$ of the real axis. Our considerations are along the lines of the classical Schur algorithm and the treatment of the Hamburger moment problem on the real axis by Nevanlinna. More precisely, a transformation of matrix sequences is constructed, which transforms Hausdorff moment sequences into Hausdorff moment sequences reduced by 1 in length. It is shown that this transformation corresponds essentially to the left shift of the associated sequences of canonical moments. As an application, we show that a matricial version of the arcsine distribution can be used to characterize a certain centrality property of non-negative Hermitian measures on $[α,β]$.

math.CA↗
arXiv2017

An Application of the Schur Complement to Truncated Matricial Power Moment Problems

The main goal of this paper is to reconsider a phenomenon which was treated in earlier work of the authors' on several truncated matricial moment problems. Using a special kind of Schur complement we obtain a more transparent insight into the nature of this phenomenon. In particular, a concrete general principle to describe it is obtained. This unifies an important aspect connected with truncated matricial moment problems.

math.CA↗
arXiv2017

Duality results for a general trigonometric approximation problem

Let $α\in(1,\infty)$ and $μ$ be a regular finite Borel measure on a locally compact abelian group. The paper deals with a general trigonometric approximation problem in $L^α(μ)$, which arises in prediction theory of harmonizable symmetric $α$-stable processes. To solve it, a duality method is applied, which is due to Nakazi and was generalized by Miamee and Pourahmadi and in the sequel successfully applied by several authors. The novelty of the present paper is that we do not make any additional assumption on $μ$. Moreover, for $α=2$, multivariate extensions are obtained.

math.FA↗
arXiv2017

Matricial Canonical Moments and Parametrization of Matricial Hausdorff Moment Sequences

In this paper we study moment sequences of matrix-valued measures on compact intervals. A complete parametrization of such sequences is obtained via a symmetric version of matricial canonical moments. Furthermore, distinguished extensions of finite moment sequences are characterized in this framework. The results are applied to the underlying matrix-valued measures, generalizing some results from the scalar theory of canonical moments.

math.CA↗
arXiv2017

On the truncated matricial Stieltjes moment problem $\mathsf{M}[[α,\infty);(s_j)_{j=0}^m,\leq]$

This paper gives via Stieltjes transform a complete description of the solution set of a matricial truncated Stieltjes-type power moment problem in the non-degenerate and degenerate cases. The approach is based on the Schur type algorithm which was worked out in the papers [arXiv:1604.07240, arXiv:1604.07629]. Furthermore, the subset of parameters is determined which corresponds to another truncated matricial Stieltjes-type moment problem.

math.CV↗
arXiv2017

On the structure of Hausdorff moment sequences of complex matrices

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.

math.CV↗
arXiv2016

A necessary condition for certain functions to preserve positive semi-definiteness on partitioned matrices

If $f$ is a symmetric complex-valued function on the $m$-fold Cartesian product of the set of non-negative reals and $A$ is a positive semi-definite $m\times m$ matrix with eigenvalues $λ_j$, we set $f(A):=f(λ_1,\dotsc,λ_m)$. It is shown that if $[f(A_{αβ})]$ is positive semi-definite whenever $[A_{αβ}]$ is a positive semi-definite matrix with positive semi-definite entries $A_{αβ}$, then $f$ has a power series expansion with positive coefficients.

math.FA↗
arXiv2016

On resolvent matrix, Dyukarev-Stieltjes parameters and orthogonal matrix polynomials via $[0,\infty)$-Stieltjes transformed sequences

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.

math.CV↗
arXiv2016

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 $α=0$. 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.

math.CV↗
arXiv2016

On a Simultaneous Approach to the Even and Odd Truncated Matricial Stieltjes Moment Problem I: An $α$-Schur-Stieltjes-type algorithm for sequences of complex matrices

The characterization of the solvability of matrix versions of truncated Stieltjes-type moment problems led to the class of $α$-Stieltjes non-negative definite sequences of complex $q \times q$ matrices. In [21], a parametrization of this class was introduced, the so-called $α$-Stieltjes parametrization. The main topic of this first part of the paper is the construction of a Schur-type algorithm which produces exactly the $α$-Stieltjes parametrization.

math.CV↗