M(P,Q)的一个小注

A small note forM(P,Q)

摘要

摘要:设 H是希尔伯特空间, P, Q是 B( H)上的正交投影.证明了 M _θ( 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 M _θ( P, Q)是到 P的距离小于等于sin θ且到Q的距离小于等于cos θ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.

Abstract:Let Hbe a Hilbert space and P, Qorthogonal projections on B( H).It is proved that M _θ( P, Q) is non-empty if and only if the dimension of the intersection of PHand ( I- Q) His the same as the dimension of the intersection of QHand ( I- P) H, where M _θ( P, Q) is the set of projections on Hwhich are within distance of sin θfrom Pand cos θfrom Q.The operator matrix decomposition form of Halmos' two projection theory is applied to prove the sufficiency.On the other hand, a linear bijection between the intersection of PHand ( I- Q) Hand the intersection of QHand ( I- P) H is constructed to give the necessity.