On the Drazin inverses involving power commutativity

We explore the Drazin inverses of bounded linear operators with power commutativity ( P Q = Q m P ) in a Hilbert space. Conditions on Drazin invertibility are formulated and shown to depend on spectral properties of the operators involved. Moreover, we prove that P ± Q is Drazin invertible if P and Q are dual power commutative ( P Q = Q m P and Q P = P n Q ) and show that the explicit representations of the Drazin inverse ( P ± Q ) D depend on the positive integers m , n ⩾ 2 .