$2\times2$分块对角算子矩阵拟谱的精细刻画

数学物理学报 ›› 2024, Vol. 44 ›› Issue (1): 12-25.

$2\times2$分块对角算子矩阵拟谱的精细刻画

申润拴(),侯国林^*()

内蒙古大学数学科学学院呼和浩特 010021

收稿日期:2022-11-30 修回日期:2023-05-15 出版日期:2024-02-26 发布日期:2024-01-10
通讯作者: 侯国林, E-mail:smshgl@imu.edu.cn
作者简介:申润拴, E-mail:65240147@qq.com
基金资助:
国家自然科学基金(12261064);国家自然科学基金(11861048);内蒙古自然科学基金(2021MS01004)

The Fine Pseudo-spectra of$2 \times 2$Diagonal Block Operator Matrices

Shen Runshuan(),Hou Guolin()

School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021

Received:2022-11-30 Revised:2023-05-15 Online:2024-02-26 Published:2024-01-10
Supported by:
National Natural Science Foundation of China(12261064);National Natural Science Foundation of China(11861048);Natural Science Foundation of Inner Mongolia(2021MS01004)

摘要/Abstract

摘要：

设$A$,$B$为可分 Hilbert 空间$X$中的稠定闭线性算子,$M_{0}=\left(\begin{array}{cc}{A} & {0}\\{0}& {B}\end{array} \right)$表示$2\times2$分块算子矩阵. 文中精细刻画算子矩阵$M_{0}$在对角扰动情形下的拟点谱、拟剩余谱与拟连续谱, 所得结论与点谱、剩余谱和连续谱的结果进行了比较, 并用例子进行了辅证. 最后, 采用空间分解技巧, 用主对角元的信息刻画$M_{0}$在上三角扰动情形下的拟点谱分布.

关键词: 算子矩阵, 拟点谱, 拟剩余谱, 拟连续谱, 拟零子空间

Abstract:

Let$A$,$B$be densely closed linear operators in a separable Hilbert space$X$and$M_{0}=\left( \begin{array} {cc}{A} & {0}\\ {0}& {B} \end{array} \right)$be the corresponding$2\times2$block operator matrices. In this paper, we establish the fine pseudo-spectra of$M_{0}$including the pseudo-point spectrum, the pseudo-residual spectrum, and the pseudo-continuous spectrum under diagonal perturbation, which are, respectively, compared with its point spectrum, residual spectrum, and continuous spectrum. And a concrete example is constructed to justify the proved result. Finally, we obtain the pseudo-point spectrum of$M_{0}$under the upper-triangular perturbation by using the technology of space decomposition.

Key words: Operator matrices, Pseudo-point spectrum, Pseudo-residual spectrum, Pseudo-continuous spectrum, Pseudo-null space

中图分类号:

O175.3

申润拴, 侯国林. $2\times2$分块对角算子矩阵拟谱的精细刻画[J]. 数学物理学报, 2024, 44(1): 12-25.

Shen Runshuan, Hou Guolin. The Fine Pseudo-spectra of$2 \times 2$Diagonal Block Operator Matrices[J]. Acta mathematica scientia,Series A, 2024, 44(1): 12-25.

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