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

$\dist (0,\sigma _{\rm e}(T)) = \sup \{ \mathcal{G}_{|\cdot|}(T)\mid |\cdot| \in \mathcal{N} \}.$

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