In this paper, we establish an analytic enclosure for the spectrum of unbounded linear operators A admitting an n×n matrix representation in a Hilbert space H = H1 ⊕ ··· ⊕ Hn. For diagonally dominant operator matrices of order 0, we show that this new enclosing set, the block numerical range Wn(A), contains the eigenvalues of A and that the approximate point spectrum of A is contained in its closure Wn(A).