Spectra of $3\times 3$ upper triangular operator matrices

Let ${H}_1$, ${H}_2$, and ${H}_3$ be complex separable Hilbert spaces. Given $A\in {B}({H}_1)$, $B\in{B}({H}_2)$, and $C\in{B} ({H}_3)$, write $M_{D,E,F}=\left(\begin{smallmatrix} A & D&E\\ 0 & B&F\\ 0&0&C \end{smallmatrix}\right)$, where $D\in {B}({H}_2,{H}_1)$, $E\in{B}({H}_3,{H}_1)$, and $F\in{B}({H}_3,{H}_2)$ are unknown operators. This paper gives a complete description of the intersection $\bigcap_{D,E,F} \sigma(M_{D,E,F})$, where $D$, $E$, and $F$ range over the respective sets of bounded linear operators. Further, we show that $\sigma(A)\cup\sigma(B)\cup\sigma(C)=\sigma(M_{D,E,F})\cup W$, where $W$ is the union of certain gaps in $\sigma(M_{D,E,F})$, which are subsets of $(\sigma(A)\cap\sigma(B))\cup(\sigma(B)\cap\sigma(C))\cup(\sigma(A) \cap\sigma(C))$. Finally, we obtain a necessary and sufficient condition for the relation $\sigma(M_{D,E,F})=\sigma(A)\cup\sigma(B)\cup\sigma(C)$ to hold for any $D$, $E$, and $F$.