The Upper Semi-Weylness and Positive Nullity for Operator Matrices

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2024-03-25 DOI:10.1007/s40840-024-01654-y

Tengjie Zhang, Xiaohong Cao, Jiong Dong

引用次数: 0

Abstract

Let H and K be infinite dimensional separable complex Hilbert spaces and B(K, H) the algebra of all bounded linear operators from K into H. Let \(A\in B(H)\) and \(B\in B(K)\). We denote by \(M_C\) the operator acting on \(H\oplus K\) of the form \(M_C=\left( \begin{array}{cc}A&{}C\\ 0&{}B\\ \end{array}\right) \). In this paper, we give necessary and sufficient conditions for \(M_C\) to be an upper semi-Fredholm operator with \(n(M_C)>0\) and \(\hbox {ind}(M_C)<0\) for some left invertible operator \(C\in B(K,H)\). Meanwhile, we discover the relationship between \(n(M_C)\) and n(A) during the exploration. And we also describe all left invertible operators \(C\in B(K,H)\) such that \(M_C\) is an upper semi-Fredholm operator with \(n(M_C)>0\) and \(\hbox {ind}(M_C)<0\).

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算子矩阵的上半完备性和正无效性

让 H 和 K 是无限维的可分离复希尔伯特空间，B(K, H) 是所有从 K 到 H 的有界线性算子的代数。我们用 \(M_C\) 表示作用于 \(H\oplus K\) 的形式为 \(M_C=\left( \begin{array}{cc}A&{}C\\0&{}B\\\end{array}\right) 的算子。）在本文中，我们给出了对于某个左可逆算子（C\in B(K,H)）来说，\(M_C\)是上半弗来霍尔算子的必要条件和充分条件，即\(n(M_C)>0\)和\(\hbox {ind}(M_C)<0\)。同时，我们在探索过程中发现了 n(M_C)和 n(A)之间的关系。我们还描述了所有的左可逆算子（C\in B(K,H)），使得\(M_C\)是一个上半弗里德霍姆算子，具有\(n(M_C)>0\)和\(\hbox {ind}(M_C)<0\)。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

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