A Blaschke-type condition for analytic functions on finitely connected domains. Applications to complex perturbations of a finite-band selfadjoint operator

TL;DR

This paper extends Blaschke-type conditions to finitely connected domains and applies these to derive Lieb-Thirring inequalities for complex perturbations of finite-band selfadjoint operators, including Jacobi matrices.

Contribution

It provides a local version of Blaschke-type conditions for finitely connected domains and applies them to obtain new Lieb-Thirring inequalities in this setting.

Findings

Derived a local Blaschke-type condition for finitely connected domains.

Established Lieb-Thirring inequalities for complex perturbations of finite-band operators.

Extended results to periodic Jacobi matrices with finite-band spectra.

Abstract

This is a sequel of a recent article by Borichev-Golinskii-Kupin, where the authors obtain Blaschke-type conditions for special classes of analytic functions in the unit disk which satisfy certain growth hypotheses. These results were applied to get Lieb-Thirring inequalities for complex compact perturbations of a selfadjoint operator with a simply connected resolvent set. The first result of the present paper is an appropriate local version of the Blaschke-type condition from Borichev-Golinskii-Kupin. We apply it to obtain a similar condition for an analytic function in a finitely connected domain of a special type. Such condition is by and large the same as a Lieb-Thirring type inequality for complex compact perturbations of a selfadjoint operator with a finite-band spectrum. A particular case of this result is the Lieb-Thirring inequality for a selfadjoint perturbation of the Schatten class of a periodic (or a finite-band) Jacobi matrix. The latter result seems to be new in such generality even in this framework.