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

TL;DR

This paper introduces an $ ext{α}$-Schur-Stieltjes-type algorithm that constructs the $ ext{α}$-Stieltjes parametrization for sequences of complex matrices, advancing the solution of matrix truncated Stieltjes moment problems.

Contribution

It develops a Schur-type algorithm specifically designed to generate the $ ext{α}$-Stieltjes parametrization for complex matrix sequences, providing a new computational tool for moment problem analysis.

Findings

The algorithm successfully produces the $ ext{α}$-Stieltjes parametrization.

It offers a constructive method for solving matrix Stieltjes moment problems.

The approach enhances understanding of the structure of $ ext{α}$-Stieltjes non-negative definite sequences.

Abstract

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