The Drazin inverse of the sum of two bounded linear operators and it’s applications

Let P and Q be bounded linear operators on a Banach space. The existence of the Drazin inverse of P+Q is proved under some assumptions, and the representations of (P+Q)D are also given. The results recover the cases P2Q = 0,QPQ = 0 studied by Yang and Liu in [19] for matrices, Q2P = 0; PQP = 0 studied by Cvetkovic and Milovanovic in [7] for operators and P2Q + QPQ = 0, P3Q = 0 studied by Shakoor, Yang and Ali in [16] for matrices. As an application, we give representations for the Drazin inverse of the operator matrix A = (ACBD).