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 $\mathcal{A}$ with unit $1$. An element $a\in\mathcal{A}$ is ag-Drazin invertible if there exists $x\in\mathcal{A}$ such that $ax=xa, \, xax=x \ {\rm and} \ a-axa\in\mathcal{A}^{acc}$, where $\mathcal{A}^{acc}\triangleq\{a\in\mathcal{A}: a-\lambda 1 \ {\rm is \ generalized \ Drazin\ invertible} \ {\rm for \ all} \ \lambda\in\mathbb{C}\backslash\{0\}\}.$ Using idempotent elements, we characterize this inverse and give some its representations. Also, we prove that $a\in\mathcal{A}$ is ag-Drazin invertible if and only if $0$ is not an accumulation point of $\sigma_{d}(a)$, where $\sigma_{d}(a)$ is the generalized Drazin spectrum of $a$.