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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Théorème de l’application spectrale pour le spectre essentiel quasi-Fredholm
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by M. Berkani and A. Ouahab PDF
Proc. Amer. Math. Soc. 125 (1997), 763-774 Request permission

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 $A=A_1\oplus A_0$ où $A_0$ 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-Fred- holm 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
  • Received by editor(s): March 13, 1995
  • Communicated by: Palle E. T. Jorgensen
  • © Copyright 1997 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 125 (1997), 763-774
  • MSC (1991): Primary 47A10, 47A53
  • DOI: https://doi.org/10.1090/S0002-9939-97-03431-X
  • MathSciNet review: 1340375