Commuting properties of generalized inverses
Michael P. Drazin · Linear and Multilinear Algebra · 2013
For any semigroup and any , in 2012, the author defined to be a -inverse of if , and . For given , any such is unique if it exists, and, by choosing and appropriately, one can arrange for to become, inter alia, either the author’s pseudo-inverse or (if is a -semigroup) the Moore–Penrose inverse . For any , the author proved in 1958 that implies whenever exists. In this article it is shown that, more generally, implies whenever and both exist. Still more generally, a corresponding result is proved for -inverses; in particular, for the Moore–Penrose inverse, and together imply . For any , new connections between the generalized invertibility of and are also obtained.