On the gz-Kato decomposition and generalization of Koliha Drazin invertibility

Abstract

In [24], Koliha proved that T ? L(X) (X is a complex Banach space) is generalized Drazin invertible operator iff there exists an operator S commuting with T such that STS = S and ?(T2S?T) ? {0} iff 0 < acc ?(T). Later, in [14, 34] the authors extended the class of generalized Drazin invertible operators and they also extended the class of pseudo-Fredholm operators introduced by Mbekhta [27] and other classes of semi-Fredholm operators. As a continuation of these works, we introduce and study the class of 1zinvertible (resp., gz-Kato) operators which generalizes the class of generalized Drazin invertible operators (resp., the class of generalized Kato-meromorphic operators introduced by Zivkovic-Zlatanovic and Duggal in [35]). Among other results, we prove that T is 1z-invertible iff T is 1z-Kato with ?p(T) = ?q(T) < ? iff there exists a commuting operator S with T such that STS = S and acc ?(T2S ? T) ? {0} iff 0 ? acc (acc ?(T)). As application and using the concept of the Weak SVEP introduced at the end of this paper, we give new characterizations of Browder-type theorems.