A spectral equivalence for Jacobi matrices

Abstract

We use the classical results of Baxter and Golinskii-Ibragimov to prove a new spectral equivalence for Jacobi matrices on l 2 (N). In particular, we consider the class of Jacobi matrices with conditionally summable parameter sequences and find necessary and sufficient conditions on the spectral measure such that ∞ k=n b k and ∞ k=n (a 2 k − 1) lie in l 2 1 ∩ l 1 or l 1 s for s 1.

On the scattering problem in Ryckman's class of Jacobi matrices

“…In 2007 Ryckman [13], [14], [15] came up with a new class of Jacobi matrices, for which he obtained a complete spectral description. To state his result we introduce some notations and definitions.…”

Spectral and resonance problem for perturbations of periodic Jacobi operators

“…These cases are rare (for a great overview, see Simon's [54,Chapt 1]). We would like to distinguish the Killip-Simon theorem for L 2 perturbations [33], Geronimo-Nevai's [21] and Ryckman's [48] papers for weighted L 1 perturbations, and Ryckman's [49] theorem for H 1/2 perturbations. Of these only the Killip-Simon theorem was generalized to the periodic setting by Damanik-Killip-Simon [10].…”