Abstract:
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.
