A matrix version of a higher-order Szeg{\"o} theorem

TL;DR

This paper extends a higher-order sum rule from scalar to matrix-valued measures on the unit circle, involving matrix Verblunsky coefficients, advancing the understanding of spectral theory in matrix analysis.

Contribution

It introduces a matrix version of a higher-order Szeg{"o} theorem, generalizing previous scalar results to matrix-valued measures and coefficients.

Findings

Established a matrix analogue of the higher-order Szeg{"o} theorem.

Connected matrix Verblunsky coefficients with spectral properties.

Extended sum rule techniques to matrix measures.

Abstract

We extend a higher-order sum rule proved by B. Simon to matrix valued measures on the unit circle and their matrix Verblunsky coefficients.