  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 $ax=xa, a-a^2x\in
A^{qnil}$. 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).
