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

Abstract

In this paper, we establish an analytic enclosure for the spectrum of unbounded linear operators~$\mathcal{A} $ admitting an $n \times n$ matrix representation in a Hilbert space $\mathcal{H} =\mathcal{H} _1\oplus \cdots \oplus \mathcal{H} _n$. For diagonally dominant operator matrices of order 0, we show that this new enclosing set, the block numerical range $W^n(\mathcal{A} )$, contains the eigenvalues of $\mathcal{A} $ and that the approximate point spectrum of $\mathcal{A} $ is contained in its closure $\overline {W^n(\mathcal{A} )}$. Since the block numerical range turns out to be a subset of the usual numerical range, $W^n(\mathcal{A} )\subset W(\mathcal{A} )$, it may give a tighter enclosure of the spectrum. Moreover, we prove Gershgorin theorems for diagonally dominant $n \times n$ operator matrices and compare our results to both Gershgorin bounds and classical perturbation theory. Our results are illustrated by deriving new lower bounds for $3\times 3$ self-adjoint operator matrices and applying the latter to three-channel Hamiltonians in quantum~mechanics.