A Note on Cline's Formula for Two Subclasses of Generalized Drazin Inverses

Abstract

For a Banach algebra $\mathcal{A}$ with identity and $a$, $b$, $c$, $d$ $∈$ $\mathcal{A}$, the relations between the extended g-Drazin inverse (resp. generalized strong Drazin inverse) of $ac$ and that of $bd$ are given, when $bac = bdb$ and $cac = cdb$.