J对称分块算子矩阵的谱  

Spectral of J-symmetric Block Operator Matrix

作　　者：钱志祥 QIAN Zhixiang(Department of Basic Education,Guangdong Polytechnic College,Zhaoqing 526100,China)

机构地区：[1]广东理工学院基础教学部,广东肇庆526100

出　　处：《四川轻化工大学学报（自然科学版）》2020年第1期67-73,共7页Journal of Sichuan University of Science & Engineering(Natural Science Edition)

基　　金：广东省教育厅自然基金项目(2017KTSCX204);广东理工学院科技项目(2019GKJZK020)。

摘　　要：基于J对称微分算子,J自伴微分算子和分块算子矩阵的定义,首先,给出了J对称分块算子矩阵和J自伴分块算子矩阵的判断定理,还给出了他们的共轭算子的性质。其次,利用分析和算子的方法,研究了J对称分块算子矩阵和J自伴分块算子矩阵的亏指数与其零空间的维数之间的关系,发现Hilbert空间上有界分块算子矩阵是J自伴的充要条件是它的亏指数等于零;再利用同样的方法,得到在Hilbert空间上的有界J自伴分块算子矩阵的剩余谱为空集的结论。Based on the definitions of J-symmetric differential operator,J-self-adjoint differential operator and block operator matrix,Firstly,The judgment theorems of J-symmetric block operator matrices and J-self-adjoint block operator matrices are given,and the properties of their conjugate operators are also given.Secondly,By using the method of analysis and operator,the relation between the deficient index of J-symmetric block operator matrix and J-self-adjoint block operator matrix and the dimension of zero space is studied.Thus,the necessary and sufficient condition for bounded block operator matrix to be J-self-adjoint on Hilbert space is obtained if its deficient index is equal to zero.Finally,Using the same method,we obtain the conclusion that the residual spectrum of the bounded J-self-adjoint block operator matrix on Hilbert space is an empty set.

关 键 词：J对称算子 J自伴算子 分块算子矩阵 剩余谱 

分 类 号：O175.3[理学—数学]