New Extensions of Cline’s Formula for Generalized Inverses

Abstract

In this paper, Cline's formula for the well-known generalized inverses such as Drazin inverse, pseudo Drazin inverse and generalized Drazin inverse is extended to the case when
$\left\{
\begin{aligned}
acd=dbd \ \\
dba=aca. \\
\end{aligned}
\right.$ Also, applications are given to some interesting Banach space operator properties like algebraic, meromorphic, polaroidness and B-Fredholmness.