Spectral inclusion for unbounded block operator matrices

In this paper we establish it new analytic enclosure for the spectrum of unbounded linear operators A admitting a block operator matrix representation. For diagonally dominant and off-diagonally dominant block operator matrices, we show that the recently introduced quadratic numerical range W-2(A) contains the eigenvalues of A and that the approximate point spectrum of A is contained in the closure of W-2(A). This provides a new method to enclose the spectrum of unbouded block operator matrices by means of the non-convex set W-2(A). Several examples illustrate that this spectral inclusion may he considerably lighter than the one by the usual numerical range or by perturbation theorems, both in the non-self-adjoint case and in the self-adjoint case. Applications to Dirac operators and to two-channel Hamiltonians are given. (C) 2009 Elsevier Inc. All rights reserved.