Right limits and reflectionless measures for CMV matrices

TL;DR

This paper investigates CMV matrices through their right-limit sets, establishing new results on the implications of absolutely continuous spectrum and extending classical asymptotic analysis of orthogonal polynomials.

Contribution

It provides a CMV matrix analogue of Remling's result on absolutely continuous spectrum and refines the understanding of spectral measure convergence using right limits.

Findings

Established a CMV version of Remling's theorem.

Extended Khrushchev's results on spectral measures and ratio asymptotics.

Reproduced Simon's results for Jacobi matrices using similar methods.

Abstract

We study CMV matrices by focusing on their right-limit sets. We prove a CMV version of a recent result of Remling dealing with the implications of the existence of absolutely continuous spectrum, and we study some of its consequences. We further demonstrate the usefulness of right limits in the study of weak asymptotic convergence of spectral measures and ratio asymptotics for orthogonal polynomials by extending and refining earlier results of Khrushchev. To demonstrate the analogy with the Jacobi case, we recover corresponding previous results of Simon using the same approach.