The generalized Drazin inverse of operator matrices

Representations for the generalized Drazin inverse of an operator matrix ((A B)(C D)) are presented in terms of A, B, C, D and the generalized Drazin inverses of A, D, under the condition that BDd = 0, and (BDC)-C-i = 0, for any nonnegative integer i. Using the representation, we give a new additive result of the generalized Drazin inverse for two bounded linear operators P, Q is an element of B(X) with PQ(d) = 0 and PQ(i)P = 0, for any integer i >= 1. As corollaries, several well-known results are generalized.