Title: Solvability of the operator Riccati equation in the Feshbach case,
Abstract: We consider a bounded block operator matrix of the form $$
L=\left(\begin{array}{cc} A & B \\ C & D \end{array} \right), $$ where the
main-diagonal entries $A$ and $D$ are self-adjoint operators on Hilbert spaces
$H_{_A}$ and $H_{_D}$, respectively; the coupling $B$ maps $H_{_D}$ to $H_{_A}$
and $C$ is an operator from $H_{_A}$ to $H_{_D}$. It is assumed that the
spectrum $\sigma_{_D}$ of $D$ is absolutely continuous and uniform, being
presented by a single band $[\alpha,\beta]\subset\mathbb{R}$, $\alpha<\beta$,
and the spectrum $\sigma_{_A}$ of $A$ is embedded into $\sigma_{_D}$, that is,
$\sigma_{_A}\subset(\alpha,\beta)$. We formulate conditions under which there
are bounded solutions to the operator Riccati equations associated with the
complexly deformed block operator matrix $L$; in such a case the deformed
operator matrix $L$ admits a block diagonalization. The same conditions also
ensure the Markus-Matsaev-type factorization of the Schur complement
$M_{_A}(z)=A-z-B(D-z)^{-1}C$ analytically continued onto the unphysical
sheet(s) of the complex $z$ plane adjacent to the band $[\alpha,\beta]$. We
prove that the operator roots of the continued Schur complement $M_{_A}$ are
explicitly expressed through the respective solutions to the deformed Riccati
equations.