Additive property of Drazin invertibility of elements in a ring

Abstract

In this article, we investigate additive properties on the Drazin inverse of elements in rings. Under the commutative condition of ab = ba, we show that a + b is Drazin invertible if and only if 1 + a D b is Drazin invertible. Not only the explicit representations of the Drazin inverse (a + b) D in terms of a, a D , b and b D , but also (1 + a D b) D is given. Further, the same property is inherited by the generalized Drazin invertibility in a Banach algebra and is extended to bounded linear operators.