The Drazin inverses of sum and difference of idempotents (Q999806)

The paper is concerned with the Drazin inverses \((P \pm Q)^{D}\) of the sum and difference of two idempotents \(P\) and \(Q\), a problem that was first considered by \textit{M. P. Drazin} in [Am. Math. Mon. J. 65, 506--514 (1958; Zbl 0083.02901)]. Using the technique of operator block matrices, the author presents some formulas for \((P \pm Q)^{D}\) in different cases: (i) \(PQP=O\); (ii) \(PQP=PQ\); (iii) \(PQP=P\).