On the perturbation functions and similarity orbits

Haïkel Skhiri¹

¹ Département de Mathématiques Faculté des Sciences de Monastir Avenue de l'environnement 5019 Monastir, Tunisie

Studia Mathematica, Tome 188 (2008) no. 1, pp. 57-66

Résumé

We show that the essential spectral radius $ \varrho_e (T)$ of $T\in B(H)$ can be calculated by the formula $ \varrho_e (T) = \inf \{ {\cal{F}}_{\sharp \cdot\sharp } (X T X^{-1}) :X$ an invertible operator$\},$ where ${\cal{F}}_{\sharp \cdot\sharp } (T)$ is a ${\mit\Phi}_1$-perturbation function introduced by Mbekhta [J. Operator Theory 51 (2004)]. Also, we show that if ${\cal{G}}_{\sharp \cdot\sharp } (T)$ is a ${\mit\Phi}_2$-perturbation function [loc. cit.] and if $T$ is a Fredholm operator, then $ \mathop{\rm dist}(0,\sigma_e(T)) = \sup \{ {\cal{G}}_{\sharp \cdot\sharp } (X T X^{-1}): X$ an invertible operator$\}.$

Détail

DOI : 10.4064/sm188-1-3

Keywords: essential spectral radius varrho calculated formula varrho inf cal sharp cdot sharp invertible operator where cal sharp cdot sharp mit phi perturbation function introduced mbekhta operator theory cal sharp cdot sharp mit phi perturbation function loc cit fredholm operator mathop dist sigma sup cal sharp cdot sharp invertible operator