The G-Drazin İnverse İnvolving Power Commutativity

Abstract

Let A be a complex Banach algebra. An element a ∈ A has g-Drazin inverse if there exists b ∈ A such that b = bab, ab = ba, a − a2b ∈ Aqnil . Let a, b ∈ Ad . If a3b = ba, b3a = ab, and a2adb = aadba, we prove that a + b ∈ Ad if and only if 1 + adb ∈ Ad . We present explicit formula for (a + b)d under certain perturbations. These extend the main results of Wang, Zhou and Chen (Filomat, 30(2016), 1185–1193) and Liu, Xu and Yu (Applied Math. Comput., 216(2010), 3652–3661). © 2020, University of Nis. All rights reserved.