Operator interpretation of resonance arising in spectral problems for 2 x 2 operator matrices

We consider operator matrices H = ((A0)(B10) (B10)(A1)) with self-adjoint e ntries A(i), i = 0, 1, and bounded B-01 = B-10* , acting in the orthogonal sum H = H-0 + H-1 of Hilbert spaces H-0 and H-1. We are especially interest ed in the case where the spectrum of, say, A(1) is partly or totally embedd ed into the continuous spectrum of Ao 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 co ntinuation (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 w here M-1(-1)(z) and consequently the resolvent (H - z)(-1) have poles are u sually called resonances. A main goal of the present work is to find non-se lfadjoint operators whose spectra include the resonances as well as to stud y the completeness and basis properties of the resonance eigenvectors of M- 1(z) in H-1. To this end we first construct an operator-valued function V-1 (Y) on the space of operators in 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 eigenval ue z and then study the equation H-1 = A(1) + V-1(H-1). We prove the solvab ility 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 t he same time such vectors for M-1(z), we prove completeness and even basis properties for the root vectors (including those for the resonances).