### A Cline's formula for the generalized Drazin-Riesz inverses

#### Abstract

Let X be Banach space, A,B,C be bounded linear operators on X satisfying operator equation ABA = ACA. In this note, we show that AC is generalized Drazin-Riesz invertible if and only if BA is generalized Drazin Riesz invertible. So, we generalize Cline’s formula to the case of the generalized Drazin-Riesz invertibility.

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