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

TL;DR

This paper develops a unified approach using Schur analysis to solve both even and odd truncated matricial Hamburger moment problems, introducing new results for the odd case and extending previous scalar and even matrix cases.

Contribution

It introduces a simultaneous treatment of even and odd matrix Hamburger moment problems using Schur-type algorithms, including a novel function-theoretic approach.

Findings

Complete description of the solution set for the moment problem.

Introduction of a new function-theoretic Schur algorithm.

Extension of results to the most general matrix case.

Abstract

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. The even case was studied earlier by G.-N. Chen and Y.-J. Hu. 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 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.