Some spectra properties of unbounded 2×2 upper triangular operator matrices

Abstract

Let 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 ${\bigcup }_{C\in {\mathcal{C}}_B^{+}(\mathcal{K},\mathcal{H})}{\sigma }_{\ast }(M_C)$ and ${\bigcap }_{C\in {\mathcal{C}}_B^{+}(\mathcal{K},\mathcal{H})}{\sigma }_{\mathrm{cr}}(M_C)$, where ${\sigma }_{\ast }(M_C)$ 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 $C:\mathcal{D}(C)\subset \mathcal{K}\to \mathcal{H}$ such that $\mathcal{D}(C)\supset \mathcal{D}(B)$ for a given closed operator $B$ acting in $\mathcal{K}$.