A new concept for block operator matrices: the quadratic numerical range

In this paper a new concept for 2 x 2-block operator matrices - the quadrat ic numerical range - is studied. The main results are a spectral inclusion theorem, an estimate of the resolvent in terms of the quadratic numerical r ange, factorization theorems for the Schur complements, and a theorem about angular operator representations of spectral invariant subspaces which imp lies e,g. the existence of solutions of the corresponding Riccati equations and a block diagonalization. All results are new in the operator as well a s in the matrix case. (C) 2001 Elsevier Science Inc. All rights reserved.