2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/169524For $a,α>0$ let $E(a,α)$ be the set of all compact operators $A$ on a separable Hilbert space such that $s_n(A)=O(\exp(-an^α))$, 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 $E(a,α)$.AMS-LaTeX, 20 pagesFunctional Analysis47A10 (Primary) 47B06, 47B07 (Secondary)text