New characterizations of g-Drazin inverse in a Banach algebra

In this paper, we present a new characterization of g-Drazin inverse in a Banach algebra. We prove that an element a in a Banach algebra has g-Drazin inverse if and only if there exists x E 51 such that ax = xa, a - a2x E 51qnil. As an application, we obtain the sufficient and necessary conditions for the existence of the g-Drazin inverse for certain 2 x 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).