On the Isolated Points of the Surjective Spectrum of a Bounded Operator

For a bounded operator T acting on a complex Banach space, we show that if T - λ is not surjective, then λ is an isolated point of the surjective spectrum $\sigma _{su}(T)$ of T if and only if X = H₀(T-λ) + K(T-λ), where H₀(T) is the quasinilpotent part of T and K(T) is the analytic core for T. More...

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Bibliographic Details
Published in:Proceedings of the American Mathematical Society 2008-10, Vol.136 (10), p.3521-3528, Article 3521
Main Authors: González, Manuel, Mbekhta, Mostafa, Oudghiri, Mourad
Format: Article
Language:English
Subjects:
Analytic functions
Banach space
Eigenvalues
Exact sciences and technology
Functional analysis
General mathematics
General, history and biography
Integers
Mathematical analysis
Mathematical theorems
Mathematics
Operator theory
Sciences and techniques of general use
Several complex variables and analytic spaces
Spectral theory
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Summary:For a bounded operator T acting on a complex Banach space, we show that if T - λ is not surjective, then λ is an isolated point of the surjective spectrum $\sigma _{su}(T)$ of T if and only if X = H₀(T-λ) + K(T-λ), where H₀(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) < ∞. We show that for each of these operators T, there exists a finite set E consisting of Riesz points for T such that 0 ∈ σ(T) \ E and σ(T) \ E is connected, and derive some consequences.
ISSN:0002-9939
1088-6826
DOI:10.1090/S0002-9939-08-09549-X