Quantitative spectral perturbation theory for compact operators on a Hilbert space

TL;DR

This paper develops a spectral perturbation theory for compact operators on Hilbert spaces, providing explicit bounds on resolvent norms and spectral distances based on operator compactness classes.

Contribution

It introduces compactness classes of Hilbert space operators and derives explicit bounds for spectral perturbations within these classes.

Findings

Derived upper bounds for resolvent norms of operators in specific compactness classes.

Established explicit bounds for the Hausdorff distance between spectra of operators in the same class.

Provided a framework linking singular value decay to spectral stability.

Abstract

We introduce compactness classes of Hilbert space operators by grouping together all operators for which the associated singular values decay at a certain speed and establish upper bounds for the norm of the resolvent of operators belonging to a particular compactness class. As a consequence we obtain explicitly computable upper bounds for the Hausdorff distance of the spectra of two operators belonging to the same compactness class in terms of the distance of the two operators in operator norm.