Commutativity of projectors Original Research Article

It is known that necessary and sufficient conditions for the sum P1+P2 and the difference P1−P2 of projectors P1 and P2 to be also projectors are P1P2=0=P2P1 and P1P2=P2=P2P1, respectively, independently of whether P1 and P2 are orthogonal or not. The situation changes when considering the products of P1 and P2: in case of orthogonal projectors the condition P1P2=P2P1 is both necessary and sufficient for P1P2 (and thus for P2P1) to be a projector, but in the general case it discontinues to be necessary even if P1P2 along with P2P1 are required to be projectors. The purpose of the present paper is to investigate similarities and dissimilarities of this kind between several results concerning orthogonal projectors and their counterparts corresponding to arbitrary projectors, with special emphasis laid on the commutativity condition. The investigations refer to matrix representations of projectors, as well as to subspaces and generalized inverses connected with them.