An extension of Hirano inverses in Banach algebras

Abstract

We introduce a new class of generalized inverse which is called ??Hirano inverse. In this paper some elementary properties of the ??Hirano inverse are obtained. We prove that a ? A is ??Hirano invertible if and only if a ? an+1 is nilpotent for some positive integer n. Certain multiplicative and additive results for the ??Hirano inverse in a Banach algebra are presented. We then apply these new results to block operator matrices over Banach spaces.