Variation of discrete spectra for non-selfadjoint perturbations of selfadjoint operators

TL;DR

This paper investigates how the discrete spectrum of a non-selfadjoint perturbation of a selfadjoint operator varies, providing bounds on spectral shifts and applications to zero-sets of Cauchy transforms.

Contribution

It establishes bounds on the sum of spectral distances for non-selfadjoint perturbations and extends these results to unitary operators and Cauchy transform zero-sets.

Findings

Bound on the sum of spectral distances in terms of Schatten norm of the perturbation

Extension of bounds to unitary operators

Application to estimates on zero-sets of Cauchy transforms

Abstract

Let B=A+K where A is a bounded selfadjoint operator and K is an element of the von Neumann-Schatten ideal S_p with p>1. Let {\lambda_n} denote an enumeration of the discrete spectrum of B. We show that $\sum_n \dist(\lambda_n, \sigma(A))^p$ is bounded from above by a constant multiple of |K|_p^p. We also derive a unitary analog of this estimate and apply it to obtain new estimates on zero-sets of Cauchy transforms.