some additive results on gs-darzin inverse

Let a be a banach algebra. an element a a has gs-drazin inverse if there existsb a such thatb = bab, ab = ba and a − ab a^qnil.additive results for the gs-drazin inverse in banach algebras are investigated. we prove thatfor any a, b a′′ , if aba^2 = 0, abab = 0, ab^2a = 0 and ab^3 = 0, then a + b a′′ . the gs-drazininvertiblity of the sum of two operators with multipacative perturbations are presented.