Nonextreme de Branges–Rovnyak Spaces as Models for Contractions

Abstract

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 ∞. 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.