Square-summable variation and absolutely continuous spectrum. (English) Zbl 1312.47038
Summary: Recent results of S. A. Denisov [Commun. Math. Phys. 226, No. 1, 205–220 (2002; Zbl 1005.34076)] and U. Kaluzhny and M. Shamis [Constr. Approx. 35, No. 1, 89–105 (2012; Zbl 1242.47024)] describe the absolutely continuous spectrum of Jacobi matrices with coefficients that obey an \(\ell^2\) 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.
MSC:
| 47B36 | Jacobi (tridiagonal) operators (matrices) and generalizations |
| 42C05 | Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis |
| 39A70 | Difference operators |
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