New extensions of Cline's formula for generalized Drazin–Riesz inverses

Abstract

In this note, Cline's formula for generalized Drazin-Riesz inverses is proved. We prove that if A, D ∈ B(X, Y) and B, C ∈ B(Y, X) are such that ACD = DBD and DBA = ACA, then AC is generalized Drazin-Riesz invertible if and only if BD is generalized Drazin-Riesz invertible, and that, in such a case, if S is a generalized Drazin-Riesz inverse of AC then T := BS 2 D is a generalized Drazin-Riesz inverse of BD.