On Invariant Graph Subspaces of a $J$-Self-Adjoint Operator in the Feshbach Case

We consider a $J$-self-adjoint $2\times2$ block operator matrix $L$ in the Feshbach spectral case, that is, in the case where the spectrum of one main-diagonal entry of $L$ is embedded into the absolutely continuous spectrum of the other main-diagonal entry. We work with the analytic continuation of the Schur complement of a main-diagonal entry in $L-z$ to the unphysical sheets of the spectral parameter $z$ plane. We present conditions under which the continued Schur complement has operator roots in the sense of Markus–Matsaev. The operator roots reproduce (parts of) the spectrum of the Schur complement, including the resonances. We, then discuss the case where there are no resonances and the associated Riccati equations have bounded solutions allowing the graph representations for the corresponding $J$-orthogonal invariant subspaces of $L$. The presentation ends with an explicitly solvable example.