Generalized Drazin invertibility of combinations of idempotents

Abstract

The paper serves as a correction to J. Math. Anal. Appl. 359 (2009) 731–738 dealing with the Drazin invertibility of combinations of idempotents p,q in a Banach algebra. As the arguments employed to do this work equally well for both the usual and generalized Drazin inverse, the latter is included in the discussion which covers the equivalence of the Drazin invertibility of 1-pq, p-q, p+q and in general of αp+βq with αβ≠0, as well as the equivalence of the Drazin invertibility of the commutator pq-qp and anticommutator pq+qp of p,q.