Spectral inclusion for unbounded diagonally dominant $n\times n$ operator matrices

We establish an analytic enclosure for the spectrum of unbounded linear operators ${\cal A}$ admitting an $n \times n$ matrix representation. For diagonally dominant operator matrices of order $0$, we show that, the block numerical range $W^n({\cal A})$, contains the eigenvalues of ${\cal A}$ and that the approximate point spectrum of ${\cal A}$ is contained in its closure $W^n({\cal A})$. Since the block numerical range turns out to be a subset of the usual numerical range $W({\cal A})$, i.e. $W^n({\cal A}) \subset W({\cal A})$,it may give a tighter enclosure of the spectrum.