具有单值扩张性的上三角算子矩阵半Fredholm性及其应用  

Semi-Fredholmness for Upper Triangular Operator Matrices with the Single-valued Extension Property and Its Application

作　　者：王晓丽 阿拉坦仓 WANG XIAOLI;ALATANCANG(School of Statistics and Mathematics,Inner Mongolia University of Finance and Economics,Hohhot 010070,China;Inner Mongolia Normal University,Hohhot 010022,China)

机构地区：[1]内蒙古财经大学统计与数学学院,呼和浩特010070 [2]内蒙古师范大学,内蒙古自治区应用数学中心,呼和浩特010022

出　　处：《应用数学学报》2023年第3期348-365,共18页Acta Mathematicae Applicatae Sinica

基　　金：国家自然科学基金资助项目(批准号:11761029);内蒙古自然科学基金项目(批准号:2018MS07020);内蒙古自治区高等学校科学技术研究项目(批准号:NJZY22323)资助项目。

摘　　要：本文研究了具有单值扩张性的上三角算子矩阵半Fredholm性,利用单值扩张性与分块算子升标、降标、零维、亏维之间的联系,得到了具有单值扩张性的上三角算子矩阵半Fredholm性刻画,给出了用对角算子刻画上三角算子矩阵半Fredholm性的条件,并研究了算子矩阵半Fredholm性的扰动问题;此外,利用所得结果研究了上三角算子矩阵的谱、本质谱和Browder谱,同时进一步考虑了Browder定理,a-Browder定理,Weyl定理,a-Weyl定理,从局部谱的角度揭示了定理之间的联系,得到了定理成立的新条件并举例验证.In this paper,we consider semi-Fredholmness for upper triangular oper-ator matrices.We obtain the characterization of semi-Fredholmness by using the re-lationship between the single-valued extension property and ascent,descent,nullity,deficiency of operators and give conditions for diagonal operators to characterize the semi-Fredholmness property.Besides,we study the perturbation of semi-Fredholmness for upper triangular operator matrices.Further,we study the spectrum,the essential spectrum and the Browder spectrum of operator matrices,give conditions for operator matrices to satisfy Browder theorem,a-Browder theorem,Weyl theorem,and a-Weyl theorem,show connection between these theorems and illustrate with examples,which generalizes the results of Fredholm theory and further reveals the deep connection be-tween Fredholm theory and local spectral theory.

关 键 词：单值扩张性 算子矩阵 半Fredholm性 局部谱 扰动 

分 类 号：O177.7[理学—数学]