Property (gab) through localized SVEP

Abstract

In this article we study the property (gab) for a bounded linear operator T ∈ L(X)\ud on a Banach space X which is a stronger variant of Browder’s theorem. We shall give several\ud characterizations of property (gab). These characterizations are obtained by using typical tools\ud from local spectral theory. We also show that property (gab) holds for large classes of operators\ud and prove the stability of property (gab) under some commuting perturbations

Property (Bv) and Tensor Product

“…3. By hypothesis T, S ∈ (J bv ), so T and S verify a-Browder's Theorem; by Theorem 4, we have that σ(T) = A σ (T) and σ(S) = A σ (S), thus by ( [25], Theorem 3.2), we obtain that T and S verify property (gaz). We have assumed that…”

“…This has been studied through the methods of the local spectral theory, through localized SV E P, under a proper closed subspace of X and also under some topological conditions and others. So, it has a lot of influence on the development of the spectral theory because the class of operators satisfying the property (Bv) is stronger than the class of operators satisfying other properties, such as those seen in [10,22,25].…”

Perturbation Theory for Property (VE) and Tensor Product

“…This article follows the same line of research as the works referenced above, but now we consider a strong variation of the Weyl-type theorems that was introduced by Sanabria et al [3,14], namely property (V E ). According to [14], if an operator T satisfies property (V E ), then T satisfies equivalently another forty-four spectral properties, among which are Weyl-type theorems such as the properties (V Π ) and (gaz) recently studied in [15,16], respectively. This arouses the interest of studying property (V E ) from different points of view.…”