三阶上三角关系矩阵的谱、点谱和两类点谱的性质

Properties of the Spectrum, Point Spectrum, and Two Types of Point Spectra of Third-Order Upper Triangular Relation Matrices

摘要: 设$H_1,H_2,H_3$ 为无穷维复可分Hilbert空间。对给定关系$A\in ℬ\mathcal{C}ℛ\left(H_1\right)$ ，$B\in ℬ\mathcal{C}ℛ\left(H_2\right)$ ，$C\in ℬ\mathcal{C}ℛ\left(H_3\right)$ ，构造三阶上三角关系矩阵$M_{D,E,F}=\left(\begin{array}{ccc}A& D& E\\ 0& B& F\\ 0& 0& C\end{array}\right)$ 。本文从值域的稠密性角度出发，对有界线性算子的点谱进行进一步分类，将其划分为两类点谱。并利用分析方法给出了三阶上三角关系矩阵$M_{D,E,F}$ 的谱，点谱和两类点谱的性质，得到了与算子矩阵不同的结果，最后用例子证明了结果的准确性。

Abstract: Let $H_1,H_2,H_3$ be infinite-dimensional complex separable Hilbert spaces. For given linear relations $A\in ℬ\mathcal{C}ℛ\left(H_1\right)$ , $B\in ℬ\mathcal{C}ℛ\left(H_2\right)$ , $C\in ℬ\mathcal{C}ℛ\left(H_3\right)$ , define the third-order upper triangular relation matrix $M_{D,E,F}=\left(\begin{array}{ccc}A& D& E\\ 0& B& F\\ 0& 0& C\end{array}\right)$ . Based on the density of the range, this paper further classifies the point spectrum of bounded linear operators into type-1 and type-2 point spectrum. Using analytical methods, the properties of the spectrum, the point spectrum, and the two types of point spectrum of the third-order upper triangular relation matrix $M_{D,E,F}$ are obtained, yielding results that differ from those for operator matrices. Finally, examples are provided to verify the correctness of the results.