摘要
设Xi是无穷维复Banach空间,L(X_(j),X_(i))是X_(j)到X_(i)上的有界线性算子全体.考虑n×n上三角算子矩阵T=(T_(ij))1≤i≤j≤n,其中T_(ij)∈L(X_(j),X_(i)),1≤i≤j≤n;T_(ij)=0,i>j.本文研究了T7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■S(T_(ii))中退化,进而给出等式S(T)=■=1S(T_(ii))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T_(ii)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.
Let X_(i) be infinite-dimensional complex Banach spaces,L(X_(j),X_(i))be the spaces of all bounded operators from X_(j) to X_(i),1≤i≤j≤n.Consider the nxn upper triangular operator matrix T=(T_(ij))_(1≤i≤j≤n),where T_(ij)∈ L(X_(j),X_(i)), 1≤i≤j≤n,T_(ij)=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 n 5(T)shrinks from ■S(T_(ii)).Further,the authors develop some sufficient conditions for i=l n the equality<S(T)=■S(T_(ii)).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 σ τ(x),and give some new conditions for the perturbation of σ(T) and σ τ(x).
作者
王晓丽
阿拉坦仓
WANG Xiaoli;Alatancang(School of Statistics and Mathematics,Inner Mongolia University of Finance and Economics,Huhhot 010070,China;Applied Mathematics Center of Inner Mongolia,Inner Mongolia Normal University,Huhhot 010022,China.)
出处
《数学年刊（A辑）》
CSCD
北大核心
2023年第1期57-70,共14页
Chinese Annals of Mathematics
基金
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)
内蒙古高等学校科学技术项目(No.NJZY22323)的资助。
关键词
单值扩张性
算子矩阵
局部谱
解析函数
扰动
Single-valued extension property
Operator matrix
Local spec-trum
Analytic function
Perturbation
