On the Pseudo Drazin Inverse of the Sum of Two Elements in a Banach Algebra

Abstract

In this paper, some additive properties of the pseudo Drazin inverse are obtained in a Banach algebra. In addition, we find some new conditions under which the pseudo Drazin inverse of the sum $a + b$ can be explicitly expressed in terms of $a, a^{\ddag}, b, b^{\ddag}$. In particular, necessary and sufficient conditions for the existence as well as the expression for  the pseudo Drazin inverse of the sum $a + b$  are obtained under certain conditions. Also, a result of Wang and Chen [Pseudo Drazin inverses in associative rings and Banach algebras, LAA 437(2012) 1332-1345] is extended.