Some additive results on Drazin inverse (Q5915462)

Drazin inverses \((P+Q)^D\) of the sums of two matrices \(P\) and \(Q\) are considered [cf. \textit{M. P. Drazin}, Am. Math. Mon. 65, 506-514 (1958; Zbl 0083.02901)]. The relation \((P+Q)^D =P^D+ Q^D\) provided \(QP=PQ=0\) is extended to the case where only the one-sided condition \(PQ=0\) is assumed. A formula for such noncommutative case is derived. Another extension is suggested for the case where only \(P^2Q=0\) is assumed.