Solvability of the Operator Riccati Equation in the Feshbach Case

Let $L$ be a bounded $2\times2$ block operator matrix whose main-diagonal entries are self-adjoint operators. It is assumed that the spectrum of one of these entries is absolutely continuous, being presented by a single finite band, and the spectrum of the other main-diagonal entry is entirely contained in this band. We establish conditions under which the operator matrix $L$ admits a complex deformation and, simultaneously, the operator Riccati equations associated with the deformed $L$ possess bounded solutions. The same conditions also ensure a Markus–Matsaev-type factorization of one of the initial Schur complements analytically continued onto the unphysical sheet(s) of the complex plane of the spectral parameter. We prove that the operator roots of this Schur complement are explicitly expressed through the respective solutions to the deformed Riccati equations.