### 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).

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