A new extension of generalized Drazin inverse in Banach algebras

Abstract

In this paper, we introduce and study a new generalized inverse, called ag-Drazin inverses in a Banach algebra A with unit 1. An element a ∈ A is ag-Drazin invertible if there exists x ∈ A such that ax = xa, xax = x and a − axa ∈ A acc , where A acc {a ∈ A : a − λ1 is generalized Drazin invertible for all λ ∈ C\{0}}. Using idempotent elements, we characterize this inverse and give some its representations. Also, we prove that a ∈ A is ag-Drazin invertible if and only if 0 is not an accumulation point of σd(a), where σd(a) is the generalized Drazin spectrum of a.