Cline's formula for g-Drazin inverses in a ring

Abstract

It is well known that for an associative ring $R$, if $ab$ has g-Drazin inverse then $ba$ has g-Drazin inverse. In this case, $(ba)^d=b((ab)^d)^2a$.
This formula is so-called Cline's formula for g-Drazin inverse, which plays an elementary role in matrix and operator theory.
In this paper, we generalize Cline's formula to the wider case. In particular, as application, we obtain new common spectral properties of bounded linear operators.