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

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<|β|, then the Szegő condition fails