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

Abstract

We explore the Drazin inverse of bounded operators with commutativity up to a factor, PQ = λ QP , 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 = PQP . Under the condition of commutativity up to a factor PQ = λ QP (resp. PQ = PQP ), we give explicit representations of the Drazin inverse ( P - Q ) D (resp. ( P + Q ) D ) in term of P , P D , Q and Q D .