Generalized Browder’s Theorem and SVEP

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 ₀(λI − T) 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.