M(P,Q)的一个小注

A small note for M P,Q

设H是希尔伯特空间,P,Q是B(H)上的正交投影.证明了Mθ(P,Q)非空的充要条件是P的值域与Q的零空间之交的维数等于P的零空间与Q的值域之交的维数,其中Mθ(P,Q)是到P的距离小于等于sinθ且到Q的距离小于等于cosθ的投影所组成的集合.首先利用Halmos提出的两个投影理论中的矩阵分解形式证明其充分性;其次通过构建PH与(I-Q)H之交到QH与(I-P)H之交的双射,证明其必要性.

Let H be a Hilbert space and P,Q orthogonal projections on B H.It is proved that MθP,Q is non-empty if and only if the dimension of the intersection of PH and(I-Q)H is the same as the dimension of the intersection of QH and(I-P)H,where MθP,Q is the set of projections on H which are within distance of sinθfrom P and cosθfrom Q.The operator matrix decomposition form of Halmos’two projection theory is applied to prove the sufficiency.On the other hand,a linear bijection bet ween the intersection of PH and(I-Q)H and the intersection of QH and(I-P)H is constructed to give the necessity.