Application of generalized inverse to the representations of idempotent operators
Hong-Ke Du · Journal of Shaanxi Normal University · 2011
By using operator block techniques,the representations of idempotent operators are discussed.It is proved that the necessary and sufficient condition for the operator A(BA)+ B on a Hilbert space to be an orthogonal projection is that PB*B=B*BP,where PB*B|R(P) is an invertible operator on R(P),and PA|R(A*B*) is an invertible operator on R(A*B*)(A+ denote the Moore-Penrose inverse of A).It is also proved that an idempotent operator P can be represented as the form A(BA)+ B if and only if PAA*=AA*P*,and an orthogonal projection P can be represented as the form A(BA)+ B if and only if PAA*=AA*P.