A Strong Szegő Theorem for Jacobi Matrices

Abstract

We use a classical result of Gollinski and Ibragimov to prove an analog of the strong Szegő theorem 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 , the linearly-weighted l 2 space.

A spectral equivalence for Jacobi matrices

“…1 A similar result is proved in [6], but with A replaced by l 2 1 . While the techniques of that paper extend to handle the case discussed here, they are quite lengthy and involved.…”

Faddeev-Marchenko scattering for CMV matrices and the Strong Szego Theorem

“…The second condition was obtain by E. Ryckman [21] who proved a counterpart of the Strong Szegö Theorem for Jacobi matrices. It corresponds to n|a n | 2 < ∞ in the CMV case and is of the form…”

Analytic Scattering Theory for Jacobi Operators and Bernstein–Szegö Asymptotics of Orthogonal Polynomials

“…Note that, in [20], necessary and sufficient conditions in terms of spectral data of H were found for V to be in the Hilbert-Schmidt class. The problem of characterization of spectral data was also solved in [24] for perturbations V in the class (it is known now as the Ryckman class) relatively close to the trace class S 1 . It looks tempting to obtain exhaustive results of such type for V ∈ S 1 .…”