Shorted算子的几何结构(英文)

Geometry Structures of Shorted Operators

使用算子分块矩阵的技巧,研究了shorted算子,揭示了任意一个正算子和它的shorted算子之间的几何结构关系.此外,对由一个自伴算子A和一个闭子空间S组成的元素对(A,S)的兼容性(compatibility)进行了研究.特别地,当A是正算子时得出了集合∏(A,S)={Q∈∏:R(Q)=S⊥,AQ=Q*A}非空的充要条件;并且对集合∏(A,S)进行了详细的刻化,这里∏和S⊥分别表示一个复Hilbert空间上的所有幂等算子构成的集合和子空间S的正交补空间.

In this note, we use the block-operator matrix technique to study the shorted operators. The relations of geometry structures between a positive operator and its shorted operator are exposed. Besides, we investigate the compatibility of the pair （A,S）, where A is a bounded linear self-adjoint operator on a complex Hilbert space H and S is a closed subspace of Y. Particularly, when A is positive, we give a detailed characterization of the set Y（A, S） = {Q ∈.R（Q） = S⊥,AQ = Q·A}, where y and S⊥ denote the set of all idempotents on H and the orthogonal complement of S, respectively.