  We consider operator matrices {\bf H}= (A_0 B_{01} \\ B_{10} A_{1}) with
self-adjoint entries A_i, i=0,1, and bounded B_{01}=B_{10}^*, acting in the
orthogonal sum {\cal H}={\cal H}_0\oplus{\cal H}_1 of Hilbert spaces {\cal H}_0
and {\cal H}_1. We are especially interested in the case where the spectrum of,
say, A_1 is partly or totally embedded into the continuous spectrum of A_0 and
the transfer function M_1(z)=A_1-z+V_1(z), where
V_1(z)=B_{10}(z-A_0)^{-1}B_{01}, admits analytic continuation (as an
operator-valued function) through the cuts along branches of the continuous
spectrum of the entry A_0 into the unphysical sheet(s) of the spectral
parameter plane. The values of z in the unphysical sheets where M_1^{-1}(z) and
consequently the resolvent (H-z)^{-1} have poles are usually called resonances.
A main goal of the present work is to find non-selfadjoint operators whose
spectra include the resonances as well as to study the completeness and basis
properties of the resonance eigenvectors of M_1(z) in {\cal H}_1. To this end
we first construct an operator-valued function V_1(Y) on the space of operators
in {\cal H}_1 possessing the property: V_1(Y)\psi_1=V_1(z)\psi_1 for any
eigenvector \psi_1 of Y corresponding to an eigenvalue z and then study the
equation H_1=A_1+V_1(H_1). We prove the solvability of this equation even in
the case where the spectra of A_0 and A_1 overlap. Using the fact that the root
vectors of the solutions H_1 are at the same time such vectors for M_1(z), we
prove completeness and even basis properties for the root vectors (including
those for the resonances).
