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In this paper, we introduce and study a new generalized inverse, calledag-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, wecharacterize this inverse and give some its representations. Also, we provethat $a\in\mathcal{A}$ is ag-Drazin invertible if and only if $0$ is not anaccumulation point of $\sigma_{d}(a)$, where $\sigma_{d}(a)$ is the generalizedDrazin spectrum of $a$.

A new extension of generalized Drazin inverse in Banach algebras