Extension of the generalized n-strong Drazin inverse

Abstract

The aim of this paper is to present an extension of the generalized n-strong Drazin inverse for Banach algebra elements using a g-Drazin invertible element rather than a quasinilpotent element in the definition of the generalized n-strong Drazin inverse. Thus, we introduce a new class of generalized inverses which is a wider class than the classes of the generalized n-strong Drazin inverse and the extended generalized strong Drazin inverses. We prove a number of characterizations for this new inverse and some of them are based on idempotents and tripotents. Several generalizations of Cline?s formula are investigated for the extension of the generalized n-strong Drazin inverse.