A Spectral Equivalence for Jacobi Matrices

Abstract

We use the classical results of Baxter and Gollinski-Ibragimov to prove a new spectral equivalence for Jacobi matrices on $l^2(\mathbb{N})$. 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 ${\sum }_{k=n}^{\infty }{b}_k$ and ${\sum }_{k=n}^{\infty }(a_k^2-1)$ lie in $l_1^2\cap l^1$ or $l_s^1$ for $s\ge 1$.

13 pages