| 王晓丽,阿拉坦仓.n × n 上三角算子矩阵的单值扩张性以及应用*[J].数学年刊A辑,2023,44(1):57~70 |
| n × n 上三角算子矩阵的单值扩张性以及应用* |
| The Single-Valued Extension Property for n × n Upper Triangular Operator Matrices and Its Application |
| Received:April 03, 2021 Revised:December 07, 2022 |
| DOI：10.16205/j.cnki.cama.2023.0005 |
| 中文关键词: 单值扩张性, 算子矩阵, 局部谱, 解析函数, 扰动 |
| 英文关键词:Single-valued extension property, Operator matrix, Local spectrum, Analytic function, Perturbation |
| 基金项目:国家自然科学基金(No.11761029), 内蒙古自治区自然科学基金重点项目(No.2022ZD05)和内蒙古高等学校科学技术项目(No.NJZY22323) |
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| 中文摘要: |
| 设Xi是无穷维复Banach空间, L(Xj,Xi)是Xj到Xi上的有界线性算子全体.考虑 n × n 上三角算子矩阵T=(Tij)1≤j≤n, 其中Tij L(Xj,Xi),1≤j≤n; Tij=0, i>j.本文研究了T的单值扩张性, 通过考察集合S(T)={λ∈C}: T在点λ没有SVEP},证明了S(T)在i=1 ? nS(Ti)中退化,进而给出等式S(T)=i=1 ? n S(Ti)成立的条件. 同时, 考察了T的单值扩张性扰动,得到了S(T)保持对角稳定时Ti所需的条件并予以证明, 同时举例说明这些条件的合理性.最后, 给出单值扩张性关于谱σ(T)和局部谱σT (x)的应用, 得到了谱扰动和局部谱扰动不变的新条件. |
| 英文摘要: |
| Let Xi be infinite-dimensional complex Banach spaces, L(Xj , Xi) be the spaces of all bounded operators from Xj to Xi , 1≤j≤n. Consider the n × n upper triangular operator matrix T = (Tij )1≤j≤n, where Tij ∈ L(Xj , Xi), 1≤j≤n, Tij = 0 for i > j. In this paper, the authors investigate the single-valued extension property for T and consider the set S(T ) = {λ ∈ C : T does not have SVEP at λ}. The authors show how S(T ) shrinks from nSi=1S(Tii). Further, the authors develop some sufficient conditions for the equality S(T ) = nS i=1 S(Tii). Also, the authors consider the perturbation for the SVEP of T and obtain some conditions. Some examples are given to illustrate these results. At the end, the authors apply the obtained results to the spectrum σ(T ) and the local spectrum σT (x), and give some new conditions for the perturbation of σ(T ) and σT (x). |
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