Nonextreme de Branges-Rovnyak spaces as models for contractions

The de Branges--Rovnyak spaces are known to provide an alternate functional model for contractions on a Hilbert space, equivalent to the Sz.-Nagy--Foias model. The scalar de Branges--Rovnyak spaces $\mathcal{H}(b)$ have essentially different properties, according to whether the defining function $b$ is or not extreme in the unit ball of $H^\infty$. For $b$ extreme the model space is just $\mathcal{H}(b)$, while for $b$ nonextreme an additional construction is required. In the present paper we identify the precise class of contractions which have as a model $\mathcal{H}(b)$ with $b$ nonextreme.Comment: 15v page