Qingmei Bai, Alatancang Chen, Junjie Huang. Source: Annals of Functional Analysis, Volume 10, Number 3, 412--424.Abstract:
Let $M_{C}=[\begin{smallmatrix}A&C\\0&B\end{smallmatrix}]:\mathcal{D}(A)\oplus \mathcal{D}(B)\subset \mathcal{H}\oplus\mathcal{K}\longrightarrow \mathcal{H}\oplus \mathcal{K}$ be a closed operator matrix acting in the Hilbert space $\mathcal{H}\oplus\mathcal{K}$ . In this paper, we concern ourselves with the completion problems of $M_{C}$ . That is, we exactly describe the sets $\bigcup _{C\in \mathcal{C}_{B}^{+}(\mathcal{K},\mathcal{H})}\sigma _{*}(M_{C})$ and $\bigcap _{C\in \mathcal{C}_{B}^{+}(\mathcal{K},\mathcal{H})}\sigma _{\mathrm{cr}}(M_{C})$ , where $\sigma _{*}(M_{C})$ includes the residual spectrum, the continuous spectrum, and the closed range spectrum of $M_{C}$ , and $\mathcal{C}_{B}^{+}(\mathcal{K},\mathcal{H})$ denotes the set of closable operators $C:\mathcal{D}(C)\subset\mathcal{K}\longrightarrow \mathcal{H}$ such that $\mathcal{D}(C)\supset \mathcal{D}(B)$ for a given closed operator $B$ acting in $\mathcal{K}$ .