上三角闭算子矩阵的本质谱和Weyl谱性质  

The Essential and Weyl Spectral Properties of Closed Upper Triangular Operator Matrices

作　　者：青梅 黄俊杰[2] 阿拉坦仓 Mei QING;Jun Jie HUANG;Alatancang(School of Mathematics and Big Data,Hohhot Minzu college,Hohhot 010051,P.R.China;School of Mathematical Sciences,Inner Mongolia University,Hohhot 010021,P.R.China;School of Mathematical Sciences,Inner Mongolia Normal University,Hohhot 010022,P.R.China)

机构地区：[1]呼和浩特民族学院数学与大数据学院,呼和浩特010051 [2]内蒙古大学数学科学学院,呼和浩特010021 [3]内蒙古师范大学数学科学学院,呼和浩特010022

出　　处：《数学学报（中文版）》2021年第5期761-772,共12页Acta Mathematica Sinica：Chinese Series

基　　金：国家自然科学基金(11961022,11761029,11961052);内蒙古自然科学基金(2018BS01005);呼和浩特民族学院博士基金(HMBS1901);呼和浩特民族学院科研创新团队建设计划内蒙古自治区本级引进高层次人才项目。

摘　　要：本文主要研究了上三角闭算子矩阵T_(B)=■的本质谱和Weyl谱的性质,其中H和K都是无穷维复可分的Hilbert空间.首先,对给定的稠定闭算子A和D,得到了存在可闭算子B使得T_(B)是半Weyl和半Fredholm算子的充分必要条件,其中B满足D(B)■D(D).进一步,刻画了T_(B)的固有本质谱和Weyl谱集合.最后,给出了等式σ_(*)(T_(B))=σ_(*)(A)∪σ_(*)(D)成立的充分必要条件,其中σ_(*)(T_(B))包含T_(B)的本质谱和Weyl谱.In this paper,the properties of essential and Weyl spectrum of the upper triangular closed operator matrix TB=■are studied,where H and K are complex infinite-dimensional separable Hilbert spaces.Fisrtly,for given closed operator A and D with dense domains,we obtain the necessary and sufficient conditions for TB to be semi-Weyl and semi-Fredholm for some closable operator B with D(B)■D(D).Moreover,the intersections of the essential and Weyl spectrum of TB are determined.Finally,the necessary and sufficient conditions for Q_(*)(TB)=σ_(*)(A)∪σ_(*)(D)are described,whereσ_(*)(TB)includes the essential and Weyl spectrum of TB.

关 键 词：闭算子 上三角算子矩阵 本质谱 WEYL谱 补问题 

分 类 号：O177.1[理学—数学] O177.7[理学—基础数学]