A note on Cline’s formula for the generalized Drazin inverse

Abstract

Liao et al. proved that if the product is generalized Drazin invertible, then so is extending the Cline’s formula to the case of the generalized Drazin invertibility. In this paper, we show that if is generalized Drazin invertible for arbitrary , then is generalized Drazin invertible in a Banach algebra. So, we generalize Cline’s formula to the case of the generalized Drazin invertibility of the powers of the products of elements and . Also, we prove that is generalized Drazin invertible if and only if is generalized Drazin invertible; and is generalized Drazin invertible if and only if is generalized Drazin invertible.