3$\times$3阶上三角型算子矩阵的可能谱

Let H1, H2 and H3 be infinite dimensional separable complex Hilbert spaces. We denote by M（D,V,F） a 3×3 upper triangular operator matrix acting on Hi ＋H2＋ H3 of theform M（D,E,F）=（A D F 0 B F 0 0 C）.For given A ∈ B（H1）, B ∈ B（H2） and C ∈ B（H3）, the sets ∪D,E,F^σp（M（D,E,F））,∪D,E,F ^σr（M（D,E,F））,∪D,E,F ^σc（M（D,E,F）） and ∪D,E,F σ（M（D,E,F）） are characterized, where D ∈ B（H2,H1）, E ∈B（H3, H1）, F ∈ B（H3,H2） and σ（·）, σp（·）, σr（·）, σc（·） denote the spectrum, the point spectrum, the residual spectrum and the continuous spectrum, respectively.

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

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.