B(H)的减偏序遗传子空间

Hereditary Subspaces With Respect to Minus Partial Order on B(H)

设B(H)是复Hilbert空间H上的有界线性算子全体,本文主要研究B(H)上的减偏序遗传子空间的特征,证明了B(H)中的弱算子拓扑闭子空间M是减偏序遗传子空间的充要条件是存在投影P,Q∈B(H),使得M=PB(H)Q.作为应用,给出了B(H)上的长度为1的初等算子保持减偏序的充要条件.

Let B（H） be the set of all bounded linear operators on a complex Hilbert space H. In this paper, we consider the structure of a hereditary subspace with respect to minus partial order in B（H）. It is proved that a weak operator topology closed subspace AA in B（H） is hereditary with respect to the minus partial order if and only if there is a pair of projections P and Q in B（H） such that M = PB（H）Q. As an application, we give a sufficient and necessary condition for an elementary operator of length one on B（H） to preserve the minus partial order.