Resolvent estimates for operators belonging to exponential classes

TL;DR

This paper establishes bounds on the resolvent norm and spectral distance for compact operators with exponentially decaying singular values, linking operator normality and spectral properties.

Contribution

It provides new upper bounds for the resolvent norm and spectral distance for operators in exponential classes, connecting decay rates to spectral stability.

Findings

Upper bounds for resolvent norms in exponential classes

Bounds on Hausdorff distance between spectra

Quantitative relation between normality and spectral stability

Abstract

For $a,\alpha>0$ let $E(a,\alpha)$ be the set of all compact operators $A$ on a separable Hilbert space such that $s_n(A)=O(\exp(-an^\alpha))$, where $s_n(A)$ denotes the $n$-th singular number of $A$. We provide upper bounds for the norm of the resolvent $(zI-A)^{-1}$ of $A$ in terms of a quantity describing the departure from normality of $A$ and the distance of $z$ to the spectrum of $A$. As a consequence we obtain upper bounds for the Hausdorff distance of the spectra of two operators in $E(a,\alpha)$.