董 炯,曹小红.算子矩阵值域的闭性及其应用[J].数学年刊A辑,2020,(4):383~298
算子矩阵值域的闭性及其应用
Closedness of Ranges for Operator Matrices and Its Application
Received:May 14, 2019  Revised:July 05, 2020
DOI:10.16205/j.cnki.cama.2020.0027
中文关键词:  值域, 半Fredholm算子, 算子矩阵, 广义Weyl算子
英文关键词:Range, Semi-Fredholm operator, Operator matrix, Generalized Weyl operator
基金项目:国家自然科学基金(No.11471200, No.11701351)和陕西省自然科学基础研究(No.2018JQ1082)
Author NameAffiliation
DONG Jiong Department of Mathematics, Changzhi University, Changzhi 046011, Shanxi,China
School of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710119, China. 
CAO Xiaohong School of Mathematics and Information Science, Shaanxi Normal University,Xi’an 710119, China. 
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中文摘要:
      令{H}和{K}均为无限复可分的Hilbert空间. 定义MX=(A&C\\X&B\)为作用在{H}}\oplus{K}上的2x2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\in{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.
英文摘要:
      Let H and K be infinite dimensional separable complex Hilbert spaces. The authors denote by MX = 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(MX) to be closed for some (or every) X ∈ B(H, K). In addition, the authors study the semi-Fredholmness and generalized Weylness of MX and give some relevant results.
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