Spectral inclusion for unbounded block operator matrices

Abstract

In this paper we establish a 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 unbounded block operator matrices by means of the non-convex set W 2 ( A ) . Several examples illustrate that this spectral inclusion may be considerably tighter 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.