The g-Drazin inverse involving 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 ? A qnil. 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).