| 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. |
