Slowly oscillating perturbations of periodic Jacobi operators in $l^{2}(\mathbb{N})$

Marcin Moszyński ¹

¹ Wydzia/l Matematyki, Informatyki i Mechaniki Uniwersytet Warszawski Banacha 2 02-097 Warszawa, Poland

Studia Mathematica, Tome 192 (2009) no. 3, pp. 259-279

Résumé

We prove that the absolutely continuous part of the periodic Jacobi operator does not change (modulo unitary equivalence) under additive perturbations by compact Jacobi operators with weights and diagonals defined in terms of the Stolz classes of slowly oscillating sequences. This result substantially generalizes many previous results, e.g., the one which can be obtained directly by the abstract trace class perturbation theorem of Kato–Rosenblum. It also generalizes several results concerning perturbations of the discrete (free or periodic) Schrödinger operator. The paper concerns “one-sided” Jacobi operators (i.e. in $ l^2({\mathbb N})$) and is based on the method of subordinacy. We provide some spectral results for the unperturbed, periodic case, and also an appendix containing some subordination theory tools.

Détail

DOI : 10.4064/sm192-3-4

Keywords: prove absolutely continuous part periodic jacobi operator does change modulo unitary equivalence under additive perturbations compact jacobi operators weights diagonals defined terms stolz classes slowly oscillating sequences result substantially generalizes many previous results which obtained directly abstract trace class perturbation theorem kato rosenblum generalizes several results concerning perturbations discrete periodic schr dinger operator paper concerns one sided jacobi operators mathbb based method subordinacy provide spectral results unperturbed periodic appendix containing subordination theory tools