, Volume 4, Issue 2, pp 215-228

Generalized Browder’s Theorem and SVEP

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Abstract.

A bounded operator \(T \in L(X), X\) a Banach space, is said to verify generalized Browder’s theorem if the set of all spectral points that do not belong to the B-Weyl’s spectrum coincides with the set of all poles of the resolvent of T, while T is said to verify generalized Weyl’s theorem if the set of all spectral points that do not belong to the B-Weyl spectrum coincides with the set of all isolated points of the spectrum which are eigenvalues. In this article we characterize the bounded linear operators T satisfying generalized Browder’s theorem, or generalized Weyl’s theorem, by means of localized SVEP, as well as by means of the quasi-nilpotent part H 0IT) as λ belongs to certain subsets of \({\mathbb{C}}\) . In the last part we give a general framework for which generalized Weyl’s theorem follows for several classes of operators.