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

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 $\lim n (a_n -1)=α$ and $\lim nb_n =β$ exist and $2α<\absβ$, then the Szegő condition fails.