A Conservative de Branges–Rovnyak Functional Model for Operator Schur Functions on $$\mathbb C^+$$ C +

We present a solution of the operator-valued Schur-function realization problem on the right-half plane by developing the corresponding de Branges–Rovnyak canonical conservative simple functional model. This model corresponds to the closely connected unitary model in the disk setting, but we work the theory out directly in the right-half plane, which allows us to exhibit structure which is absent in the disk case. A main feature of the study is that the connecting operator is unbounded, and so we need to make use of the theory of well-posed continuous-time systems. In order to strengthen the classical uniqueness result (which states uniqueness up to unitary similarity), we introduce non-invertible intertwinements of system nodes.