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 $ax=xa,xax=x\mathrm{and}a-axa\in {\mathcal{A}}^{acc}$, where ${\mathcal{A}}^{acc}\triangleq \{a\in \mathcal{A}:a-\lambda 1\mathrm{is}\mathrm{generalized}\mathrm{Drazin}\mathrm{invertible}\mathrm{for}\mathrm{all}\lambda \in \mathbb{C}\backslash \{0\}\}.$ 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 $a$.