Resolvent estimates for operators belonging to exponential classes

Abstract

For $a,\alpha >0$ let $E(a,\alpha )$ be the set of all compact operators $A$ on a separable Hilbert space such that $s_n(A)=O(\exp (-a{n}^{\alpha }))$, 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 $E(a,\alpha )$.

AMS-LaTeX, 20 pages