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##### 1936-1944
New extensions of Cline's formula for generalized Drazin–Riesz inverses
Volume 60, no. 2 (2019), pp. 567–572

### Abstract

In this note, Cline's formula for generalized Drazin–Riesz inverses is proved. We prove that if $A, D \in \mathcal{B}(X, Y)$ and $B, C \in \mathcal{B} (Y, X)$ 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 $T := BS^2D$ is a generalized Drazin–Riesz inverse of $BD$.