Square-summable variation and absolutely continuous spectrum

TL;DR

This paper extends results on the absolutely continuous spectrum from Jacobi matrices to orthogonal polynomials on the unit circle, relaxing conditions on coefficient variation and asymptotic behavior.

Contribution

It generalizes previous spectral results to the unit circle case and weakens the conditions needed for the convergence of coefficients.

Findings

Extended spectral results to orthogonal polynomials on the unit circle.

Relaxed conditions for convergence to an isospectral torus.

Removed asymptotic periodicity requirement for p=1 and p=2.

Abstract

Recent results of Denisov and Kaluzhny-Shamis describe the absolutely continuous spectrum of Jacobi matrices with coefficients that obey an l^2 bounded variation condition with step p and are asymptotically periodic. We extend these results to orthogonal polynomials on the unit circle. We also replace the asymptotic periodicity condition by the weaker condition of convergence to an isospectral torus and, for p=1 and p=2, we remove even that condition.