ON THE GENERALIZED DRAZIN INVERSE IN A BANACH ALGEBRA

generalized Drazin inverse

Abstract

We give the representations of (a + b)^d, (a + ab)^d and (a + ba)^d in terms of a, b, a^d and b^d being elements of a Banach algebra with a³b = ba and b³a = ab.  We also give the representations of (a + b)^d under the assumptions a³b = ba, \(\parallel\)a^Db\(\parallel\) < 1, and \(\parallel\)(1− aa^D)b^Da\(\parallel\) < 1.