Square-summable variation and absolutely continuous spectrum

Abstract

Recent results of Denisov [5] and Kaluzhny–Shamis [9] describe the absolutely continuous spectrum of Jacobi matrices with coefficients that obey an $\mathcal l^2$ 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.