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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


Théorème de l'application spectrale
pour le spectre essentiel quasi-Fredholm

Authors: M. Berkani and A. Ouahab
Journal: Proc. Amer. Math. Soc. 125 (1997), 763-774
MSC (1991): Primary 47A10, 47A53
MathSciNet review: 1340375
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Abstract | References | Similar Articles | Additional Information

Abstract: In 1958, T. Kato proved that a closed semi-Fredholm operator $A$ in a Banach space can be written $A=A_1\oplus A_0$ where $A_0$ is a nilpotent operator and $A_1$ is a regular one.

J. P. Labrousse studied and characterised this class of operators in the case of Hilbert spaces. He also defined a new spectrum named ``essential quasi-Fredholm spectrum'' and denoted $\sigma _e(A)$.

In this paper we prove that the essential quasi-Fredholm spectrum defined by J. P. Labrousse satisfies the mapping spectral theorem, i.e.: If $A$ is a bounded operator in a Hilbert space and $f$ an analytic function in a neighbourhood of the spectrum $\sigma (A)$ of $A$, then $f(\sigma _e(A)) =\sigma _e(f(A))$.

RÉSUMÉ. En 1958, T. Kato a montré que si $A$ est un opérateur fermé dans un espace de Banach et semi-Fredholm, alors il existe $A_1,A_0$ tels que $A=A_1\oplus A_0$$A_0$ est nilpotent et $A_1$ est régulier.

J. P. Labrousse a étudié et caractérisé cette classe d'opérateurs dans le cadre des espaces de Hilbert et a défini un nouveau spectre qu'on appelle ``spectre essentiel quasi-Fredholm'' et noté par $\sigma _e(A)$.

Dans ce travail nous allons démontrer que le spectre essentiel quasi-Fredholm défini par J. P. Labrousse vérifie le théorème de l'application spectrale, c'est à dire: Si $A$ est un opérateur bourné d'un espace de Hilbert dans lui même et $f$ une fonction analytique au voisinage du spectre $\sigma (A)$ de $A$, alors $f(\sigma _e(A))=\sigma _e(f(A))$.

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Additional Information

M. Berkani
Affiliation: International Centre for Theoretical Physics, Trieste, Italy
Address at time of publication: Département de Mathématiques, Faculté des Sciences, Université Mohammed I, Oujda, Morocco

A. Ouahab
Affiliation: Department of Mathematics, Faculty of Sciences, Université Mohammed I, Oujda, Morocco

PII: S 0002-9939(97)03431-X
Keywords: Op\'erateur r\'egulier, semi-Fredholm, quasi-Fredholm, spectre essentiel
Received by editor(s): March 13, 1995
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1997 American Mathematical Society

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