Proceedings of the American Mathematical Society

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



On the isolated points of the surjective spectrum of a bounded operator

Authors: Manuel González, Mostafa Mbekhta and Mourad Oudghiri
Journal: Proc. Amer. Math. Soc. 136 (2008), 3521-3528
MSC (2000): Primary 47A53; Secondary 47A68, 46B04
Published electronically: May 15, 2008
MathSciNet review: 2415036
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Abstract: For a bounded operator $ T$ acting on a complex Banach space, we show that if $ T-\lambda$ is not surjective, then $ \lambda$ is an isolated point of the surjective spectrum $ \sigma_{su}(T)$ of $ T$ if and only if $ X=H_0(T-\lambda)+K(T-\lambda)$, where $ H_0(T)$ is the quasinilpotent part of $ T$ and $ K(T)$ is the analytic core for $ T$. Moreover, we study the operators for which $ \dim K(T) < \infty$. We show that for each of these operators $ T$, there exists a finite set $ E$ consisting of Riesz points for $ T$ such that $ 0\in \sigma (T)\setminus E$ and $ \sigma (T)\setminus E$ is connected, and derive some consequences.

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Additional Information

Manuel González
Affiliation: Departamento de Matemáticas, Facultad de Ciencias, Universidad de Cantabria, E-39071 Santander, España

Mostafa Mbekhta
Affiliation: Université de Lille I, UFR de Mathématiques, 59655 Villeneuve d’Ascq cedex, France

Mourad Oudghiri
Affiliation: Département de Mathématiques et Informatique, Faculté des Sciences d’Oujda, Maroc

Keywords: Isolated points of the surjective spectrum, analytic core, quasinilpotent part of an operator.
Received by editor(s): July 2, 2007
Published electronically: May 15, 2008
Additional Notes: This research was partially supported by DGI (Spain), Proyecto MTM2007-67994.
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2008 American Mathematical Society