Journal of Operator Theory

Volume 51, Issue 1, Winter 2004 pp. 3-18.

Fonctions perturbation et formules du rayon spectral essentiel et de distance au spectre essentiel

Authors: Mostafa Mbekhta
Author institution: Universite de Lille I, UFR de Math\'ematiques, UMR-CNRS 8524, F-59655 Villeneuve d'Ascq, France

Summary: Let ($X, \| \cdot\| $) be a Banach space. Let $\mathcal{N}$ be the set of norms on $X$ that are equivalent with $\| \cdot\| $. For $T\in B(X)$ and $| \cdot | \in \mathcal{N}$, we introduce a Fredholm perturbation function $\mathcal{F}_{|\cdot|}(T)$ (for example: the measure of non-compactness; the essential norm, etc.). In this article we show that $r_{\rm e}(T)$, the essential spectral radius of $T$, can be calculated by the following formula: $r_{\rm e}(T) = \inf \{\mathcal{F}_{|\cdot|}(T)\mid |\cdot| \in \mathcal{N} \}.$ In addition, we introduce another perturbation function $\mathcal{G}_{|\cdot|}(T)$ (for example: the essential conorm, the distance with respect to the set of Fredholm operators, etc.) and we show that if $T$ is Fredholm then $\dist (0,\sigma _{\rm e}(T)) = \sup \{ \mathcal{G}_{|\cdot|}(T)\mid |\cdot| \in \mathcal{N} \}.$

Keywords: Essential spectrum, essential spectral radius, Fredholm operator

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