The Drazin inverse of bounded operators with commutativity up to a factor

We explore the Drazin inverse of bounded operators with commutativity up to a factor, PQ=λQPPQ=λQP, in a Banach space. Conditions on Drazin invertibility are formulated and shown to depend on spectral properties of the operators involved. We also present a result concerning the more general problem of commutativity up to a related operator factor, PQ=PQPPQ=PQP. Under the condition of commutativity up to a factor PQ=λQPPQ=λQP (resp. PQ=PQPPQ=PQP), we give explicit representations of the Drazin inverse (P-Q)D(P-Q)D (resp. (P+Q)D(P+Q)D) in term of P,PD,QP,PD,Q and QDQD.