算子矩阵值域的闭性及其应用

Closedness of Ranges for Operator Matrices and Its Application

令H和K均为无限复可分的Hilbert空间.定义M_(X)=(A C X B)为作用在H⊕K上的2×2算子矩阵,其中X为从H到K上未知的有界线性算子.在本文中,基于R(C)的闭性对某个(或任意的)X∈B(H,K),使得R(M_(X))为闭集的充要条件做了等价刻画.另外,研究了算子矩阵M_(X)的半Fredholm性与广义Weyl性并给出了一些相应的结论.

Let H and K be infinite dimensional separable complex Hilbert spaces.The authors denote byM_(X)=(A C X B)a 2×2 operator matrix acting on H⊕K,where X is an unknown bounded linear operator from H to K.In this paper,based on the closedness of R(C),the authors characterize the necessary and sufficient condition for R(M_(X))to be closed for some(or every)X∈B(H,K).In addition,the authors study the semi-Fredholmness and generalized Weylness of M_(X)and give some relevant results.