Sum Rules and the Szego Condition for Orthogonal Polynomials on the Real Line

TL;DR

This paper investigates the validity of sum rules for Jacobi matrices, extending classical results and establishing conditions under which the Szeg\

Contribution

It provides new criteria for the validity of sum rules and extends Shohat's theorem to cases with infinite point spectrum, advancing spectral theory.

Findings

Sum rule validity depends on specific spectral conditions.

Extension of Shohat's theorem to infinite spectrum cases.

Identification of conditions where Szeg\

Abstract

We study the Case sum rules, especially $C_0$, for general Jacobi matrices. We establish situations where the sum rule is valid. Applications include an extension of Shohat's theorem to cases with an infinite point spectrum and a proof that if $\lim n (a_n -1)=\alpha$ and $\lim nb_n =\beta$ exist and $2\alpha <\abs{\beta}$, then the Szeg\H{o} condition fails.