The Moore-Penrose inverse of differences and products of projectors in a   ring with involution
  In this paper, we study the Moore-Penrose inverses of differences and products of projectors in a ring with involution. Also, some necessary and sufficient conditions for the existence of such inverses are given, and their expressions are presented. 
Huihui Zhu, Jianlong Chen, Pedro Patricio
math.RA