广义逆在幂等算子表示中的应用

Application of generalized inverse to the representations of idempotent operators

利用算子分块方法讨论了使用广义逆表示幂等算子的问题.证明了Hilbert空间上幂等算子A(BA)+B成为正交投影的充要条件是PB*B=B*BP(这里A+表示A的Moore-Penrose逆),其中PB*B|R(P)是R(P)上的可逆算子,PA|R(A*B*)是R(A*B*)上的可逆算子.得出幂等算子P能表示成形如A(BA)1B的形式当且仅当PAA*=AA*P*,正交投影P能表示成形如A(BA)+B的形式当且仅当PAA*=AA*P.

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.