Generalized Hirano inverses in Banach algebras

Abstract

Let $\mathcal{A}$ be a Banach algebra. An element $a\in \mathcal{A}$ has generalized Hirano inverse if there exists $b\in \mathcal{A}$ such that $$b=bab, ab=ba, a^2-ab\in \mathcal{A}^{qnil}.$$ We prove that $a\in \mathcal{A}$ has generalized Hirano inverse if and only if $a$ has g-Drazin inverse and $a-a^3\in \mathcal{A}^{qnil}$, if and only if $a$ is the sum of a tripotent and a quasinilpotent that commute. The Cline's formula for generalized Hirano inverses are thereby obtained. Let $a,b\in \mathcal{A}$ have generalized Hirano inverses. If $a^2b=aba$ and $b^2a=bab$, we prove that $a+b$ has generalized Hirano inverse if and only if $1+a^db$ has generalized Hirano inverse. Hirano inverses of operator matrices on Banach spaces are also studied.