Bi-Inner Dilations and Bi-Stable Passive Scattering Realizations of Schur Class Operator-Valued Functions

Abstract

Let S(U; Y) be the class of all Schur functions (analytic contractive functions) whose values are bounded linear operators mapping one separable Hilbert space U into another separable Hilbert space Y , and which are defined on a domain \(\Omega \subset {\mathbb{C}}\), which is either the open unit disk \({\mathbb{D}}\) or the open right half-plane \({\mathbb{C}}^+\). In the development of the Darlington method for passive linear time-invariant input/state/output systems (by Arov, Dewilde, Douglas and Helton) the following question arose: do there exist simple necessary and sufficient conditions under which a function \(\theta \in S(U; Y )\) has a bi-inner dilation \(\Theta = \left[ \begin{array}{ll} \theta _{11}&\theta \\ \theta _{21}&\theta _{22} \end{array} \right] \) mapping \(U_1 \bigoplus U\) into \(Y \bigoplus Y1\); here U1 and Y1 are two more separable Hilbert spaces, and the requirement that Θ is bi-inner means that Θ is analytic and contractive on Ω and has unitary nontangential limits a.e. on ∂Ω. There is an obvious well-known necessary condition: there must exist two functions \(\psi_r \in S(U; Y_{1})\) and \(\psi_l \in S(U_{1}; Y)\) (namely \(\psi_r = \theta_{22}\) and \(\psi_l = \theta_{11}\)) satisfying \(\psi_{r}^{*}(z)\psi_{r}(z) = I - \theta^{*}(z)\theta(z)\) and \(\psi_{l}(z)\psi_{l}^*(z) = I - \theta(z)\theta^{*}(z)\) for almost all \(z \in \partial\Omega\). We prove that this necessary condition is also sufficient. Our proof is based on the following facts. 1) A solution ψr of the first factorization problem mentioned above exists if and only if the minimal optimal passive realization of θ is strongly stable. 2) A solution ψl of the second factorization problem exists if and only if the minimal *-optimal passive realization of θ is strongly co-stable (the adjoint is strongly stable). 3) The full problem has a solution if and only if the balanced minimal passive realization of θ is strongly bi-stable (both strongly stable and strongly co-stable). This result seems to be new even in the case where θ is scalar-valued.