A NEW CHARACTERIZATION OF G-DRAZIN INVERSE AND ITS APPLICATIONS

Abstract

In this paper, we present a new characterization
of g-Drazin inverse in a Banach algebra. We prove that an
element a is a Banach algebra has g-Drazin inverse if and only
if there exists x\in A such that ax = xa; a - a^2x\in Aqnil:
As application, we obtain the sufficient and necessary conditions
for the existence of the g-Drain inverse for certain 2 \times 2
anti-triangular matrices over a Banach algebra. These extend
the results of Koliha (Glasgow Math. J., 38(1996), 367–381),
Nicholson (Comm. Algebra, 27(1999), 3583–3592 and Zou et
al. (Studia Scient. Math. Hungar., 54(2017), 489–508).