The representations of the g-Drazin inverse in a Banach algebra

The aim of this paper is to establish an explicit representation of the generalized Drazin inverse $insert ignore into journalissuearticles values(a+b);^d$ under the condition $$ab^2=0, ba^2=0, a^{\pi}b^{\pi}insert ignore into journalissuearticles values(ba);^2=0.$$ Furthermore, we apply our results to give some representation of generalized Drazin inverse for a $2\times 2$ block operator matrix. These extend the results on Drazin inverse of Bu, Feng and Bai [Appl. Math. Comput. 218, 10226-10237, 2012] and Dopazo and Martinez-Serano [Linear Algebra Appl. 432, 1896-1904, 2010].