Further results on Left and Right Generalized Drazin Invertible Operators

Abstract

In this paper we present some new characteristics and expressions of left and right generalized Drazin invertible bounded operators on a Banach space $X.$ An explicit formula relating the left and the right generalized Drazin inverses to spectral idempotents is provided. In addition, we give a characterization of operators in $\mathcal{B}_{l}(X)$ (resp. $\mathcal{B}_{r}(X)$) with equal spectral idempotents, where $\mathcal{B}_{l}(X)$ (resp. $\mathcal{B}_{r}(X)$) denotes the set of all left (resp. right) generalized Drazin invertible bounded operators on $X.$ Next, we give some sufficient conditions which ensure that the product of elements of $\mathcal{B}_{l}(X)$ (resp. $\mathcal{B}_{r}(X)$) remains in $\mathcal{B}_{l}(X)$ (resp. $\mathcal{B}_{r}(X)$). Finally, we extend Jacobson's lemma for left and right generalized Drazin invertibility. The provided results extend certain earlier works given in the literature.