Schur analysis of matricial Hausdorff moment sequences

TL;DR

This paper extends classical moment problem algorithms to the matrix case, providing a transformation that simplifies matricial Hausdorff moment sequences and linking them to canonical moments, with applications to matrix distributions.

Contribution

It introduces a matrix-based Schur algorithm for the Hausdorff moment problem, generalizing classical scalar methods and connecting to matrix canonical moments.

Findings

Transformation reduces sequence length by 1

Links to matrix canonical moments

Characterizes measures via matricial arcsine distribution

Abstract

We develop the algebraic instance of an algorithmic approach to the matricial Hausdorff moment problem on a compact interval $[\alpha,\beta]$ 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 $[\alpha,\beta]$.