On generalized sum rules for Jacobi matrices

TL;DR

This paper extends sum rule identities for Jacobi matrices using functional analysis and Szeg"o Theorem, leading to new asymptotic results for orthonormal polynomials.

Contribution

It generalizes sum rule identities for Jacobi matrices and applies these to derive novel asymptotics for orthonormal polynomials.

Findings

Generalized sum rule identities for Jacobi matrices

New asymptotic formulas for orthonormal polynomials

Application of classical Szeg"o Theorem in a broader context

Abstract

This work is in a stream initiated by a paper of Killip and Simon [Ann. of Math. (2003)]. Using methods of Functional Analysis and the classical Szeg\"o Theorem we prove sum rule identities in a very general form. Then, we apply the result to obtain new asymptotics for orthonormal polynomials.