Maximal solutions for embedding problem for a generalized Schur function and optimal dissipative scattering systems with Pontryagin state spaces (Q2784968)

Let \(U,Y\) be separable Hilbert spaces and let \(S_k(U,Y)\) be the generalized Schur class of meromorphic operator-valued functions \(\theta\) on the unit disk with values in \(L(U,Y)\) such that the kernels NEWLINE\[NEWLINE K_\theta (z,w)=(1-\overline{w}z)^{-1}(I-\theta^*(w)\theta(z)) NEWLINE\]NEWLINE have, in a certain sense, \(k\) negative squares. The authors consider the problem of finding, for a given \(\theta \in S_k(U,Y)\), a function \(\varphi\) such that \(\left[\begin{smallmatrix} \theta \\ \varphi \end{smallmatrix}\right]\in S_k(U,Y\oplus Y_\varphi)\), where \(Y_\varphi\) is another Hilbert space. The set of all solutions \(\varphi\) is described and the existence of a maximal solution is proved. The latter is shown to be connected with the construction of an optimal dissipative scattering system with the transfer function \(\theta\).