3$\times$3阶上三角型算子矩阵的可能谱

Possible Spectrums of 3$\times$3 Upper Triangular Operator Matrices

中文关键词:  3$\times$3阶上三角型算子矩阵  点谱  剩余谱  连续谱  谱.

英文关键词:3$\times$3 upper triangular operator matrices  point spectrum  continuous spectrum  residual spectrum  spectrum.

中文摘要:

      设$H_{1}$, $H_{2}$ 和$H_{3}$是无穷维复可分Hilbert空间, $M_{(D, E, F)}$表示$H_{1}\oplus H_{2}\oplus H_{3}$上的3$\times$3阶上三角型算子矩阵$M_{(D, E, F)}\!\!=\!\!\left(%\begin{array}{ccc} A & D & E \\ 0 & B & F \\ 0 & 0 & C \\ \end{array}% \right)$. 对给定的$A\in{\mathcal{B}}(H_{1})$, $B\in{\mathcal{B}}(H_{2})$和$C\in{\mathcal{B}}(H_{3})$, 描述了集合$\bigcup\limits_{D, E, F}\!\!\!\sigma_{p}(M_{(D, E,F)})$, $\bigcup\limits_{D, E, F}\!\!\!\sigma_{r}(M_{(D, E, F)})$, $\bigcup\limits_{D, E, F}\!\!\!\sigma_{c}(M_{(D, E,F)})$和$\bigcup\limits_{D, E, F}\!\!\!\sigma(M_{(D, E, F)})$, 其中$D\in{\mathcal{B}}(H_{2},H_{1})$, $E\in {\mathcal{B}}(H_{3},H_{1})$, $F\in {\mathcal{B}}(H_{3},H_{2})$并且$\sigma(\cdot)$, $\sigma_{p}(\cdot)$, $\sigma_{r}(\cdot)$, $\sigma_{c}(\cdot)$分别表示谱, 点谱, 剩余谱与连续谱.

英文摘要:

      Let $H_{1}$, $H_{2}$ and $H_{3}$ be infinite dimensional separable complex Hilbert spaces. We denote by $M_{(D, E, F)}$ a 3$\times$3 upper triangular operator matrix acting on $H_{1}\oplus H_{2}\oplus H_{3}$ of the form $M_{(D, E, F)}\!\!=\!\!\left(%\begin{array}{ccc} A & D & E \\ 0 & B & F \\ 0 & 0 & C \\\end{array}%\right)$. For given $A\in{\mathcal{B}}(H_{1})$,$B\in{\mathcal{B}}(H_{2})$ and $C\in{\mathcal{B}}(H_{3})$, the sets$\bigcup_{D, E, F}\sigma_{p}(M_{(D, E, F)})$,$\bigcup_{D, E, F}\sigma_{r}(M_{(D, E, F)})$,$\bigcup_{D, E, F}\sigma_{c}(M_{(D, E, F)})$ and $\bigcup_{D, E, F}\sigma(M_{(D, E, F)})$ are characterized, where $D\in{\mathcal{B}}(H_{2},H_{1})$, $E\in {\mathcal{B}}(H_{3},H_{1})$, $F\in{\mathcal{B}}(H_{3},H_{2})$ and $\sigma(\cdot)$, $\sigma_{p}(\cdot)$, $\sigma_{r}(\cdot)$, $\sigma_{c}(\cdot)$ denote the spectrum, the point spectrum, the residual spectrum and the continuous spectrum, respectively.