Matrix measures on the unit circle, moment spaces, orthogonal polynomials and the Geronimus relations

TL;DR

This paper explores the structure of matrix measures on the unit circle, characterizes their moment space, and connects orthogonal polynomials, canonical moments, and Verblunsky coefficients, providing new proofs of classical relations.

Contribution

It introduces a geometric perspective on canonical moments, links them to Verblunsky coefficients, and offers an alternative proof of the Geronimus relations for matrix measures.

Findings

Characterization of moment points via block Toeplitz matrices

Explicit representation of orthogonal polynomials for matrix measures

Equivalence of geometric canonical moments and Verblunsky coefficients

Abstract

We study the moment space corresponding to matrix measures on the unit circle. Moment points are characterized by non-negative definiteness of block Toeplitz matrices. This characterization is used to derive an explicit representation of orthogonal polynomials with respect to matrix measures on the unit circle and to present a geometric definition of canonical moments. It is demonstrated that these geometrically defined quantities coincide with the Verblunsky coefficients, which appear in the Szeg\"{o} recursions for the matrix orthogonal polynomials. Finally, we provide an alternative proof of the Geronimus relations which is based on a simple relation between canonical moments of matrix measures on the interval [-1,1] and the Verblunsky coefficients corresponding to matrix measures on the unit circle.