The Representations of the G-Drazin Inverse in a Banach Algebra

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