Operator Interpretation of Resonances Arising in Spectral Problems for 2 × 2 Operator Matrices

We consider operator matrices \documentclass{article}\pagestyle{empty}\begin{document}${\rm{H = }}\left( {{_{B_{10} }^{A{_0}}} {_{A_{1} }^{B{_{01}}}} } \right)$\end{document} with self ‐ adjoint entries A i , i = 0,1, and bounded B 10 = B 10 acting in the orthogonal sum μ=μ 0 ⊕ μ 1 of Hilbert spaces μ 0 and μ 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 V 1 ( z ) = B 10 where (χ ‐ 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 χ 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 μ 1 . To this end we first construct an operator ‐ valued function V 1 ( Y ) on the space of operators in μ 1 possessing the property: V 1 ( Y )ϕ 1 = V 1 ( z )ϕ 1 for any eigenvector ϕ 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).