Properties of one-sided generalized Drazin inverses in Banach algebras

Abstract

New characterizations of left and right generalized Drazin inverses are presented in Banach algebra using tripotents. We explore Cline’s formula for those inverses. Also, we find the conditions under which the product of two left (or right) generalized Drazin invertible elements has the appropriate one-sided generalized Drazin inverse. Consequently, we study when the sum of two zero product elements has a one-sided generalized Drazin inverse. Perturbation results for the left (or right) generalized Drazin inverse are also investigated. Applying our results, we obtain new properties of the left (or right) Drazin inverse and recover known results about the generalized Drazin inverse.