Fredholm complements of upper triangular operator matrices

IF 1.1 2区 数学 Q1 MATHEMATICS
Sinan Qiu, Lining Jiang
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引用次数: 0

Abstract

For a given operator pair \((A,B)\in (B(H),B(K))\), we denote by \(M_C\) the operator acting on a complex infinite dimensional separable Hilbert space \(H\oplus K\) of the form \(M_C=\bigl ( {\begin{matrix} A&{}C\\ 0&{}B \\ \end{matrix}}\bigr )\). This paper focuses on the Fredholm complement problems of \(M_C\). Namely, via the operator pair (AB), we look for an operator \(C\in B(K,H)\) such that \(M_C\) is Fredholm of finite ascent with nonzero nullity. As an application, we initiate the concept of the property (C) as a variant of Weyl’s theorem. At last, the stability of property (C) for \(2\times 2\) upper triangular operator matrices is investigated by the virtue of the so-called entanglement spectra of the operator pair (AB).

上三角算子矩阵的弗雷德霍尔补集
对于给定的算子对 ((A,B)\in (B(H),B(K))),我们用 \(M_C\) 表示作用于复数无限维可分离希尔伯特空间 \(H\oplus K\) 的算子,其形式为 \(M_C=\bigl ( {\begin{matrix} A&{}C\\0&{}B \\end{matrix}}\bigr )\).本文的重点是 \(M_C\) 的弗雷德霍姆补全问题。也就是说,通过算子对(A, B),我们寻找一个算子 \(C\in B(K,H)\) 使得 \(M_C\) 是具有非零无效性的有限上升的弗雷德霍姆算子。作为应用,我们提出了作为韦尔定理变体的性质(C)的概念。最后,我们利用算子对(A, B)的所谓纠缠谱研究了 \(2\times 2\) 上三角算子矩阵的性质(C)的稳定性。
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
CiteScore
2.00
自引率
8.30%
发文量
67
审稿时长
>12 weeks
期刊介绍: The Banach Journal of Mathematical Analysis (Banach J. Math. Anal.) is published by Birkhäuser on behalf of the Tusi Mathematical Research Group. Banach J. Math. Anal. is a peer-reviewed electronic journal publishing papers of high standards with deep results, new ideas, profound impact, and significant implications in all areas of functional analysis and operator theory and all modern related topics. Banach J. Math. Anal. normally publishes survey articles and original research papers numbering 15 pages or more in the journal’s style. Shorter papers may be submitted to the Annals of Functional Analysis or Advances in Operator Theory.
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