Generalized Drazin invertibility of the product and sum of two elements in a Banach algebra and its applications

Abstract

Let $a,b$ be two commutative generalized Drazin invertible elements in a Banach algebra; the expressions for the generalized Drazin inverse of the product $ab$ and the sum $a+b$ were studied in some current literature on this subject. In this paper, we generalize these results under the weaker conditions $a^{2}b=aba$ and $b^{2}a=bab$. As an application of our results, we obtain some new representations for the generalized Drazin inverse of a block matrix with the generalized Schur complement being generalized Drazin invertible in a Banach algebra, extending some recent works.