A new functional model for contractions

Abstract: The paper presents a new functional model for completely non-unitary contractions on a Hilbert space. This model is based on the observation that the theory of contractions shares a common geometric basis with the extension theory of symmetric operators recently developed by the author in \cite{wang2024complex}. Compared with the now classical Sz.-Nagy-Foias model and the de Branges-Rovnyak model, ours is intrinsic in the sense that we need not construct a bigger space $H$ including the model space $\mathfrak{H}$ and realize the model operator on $\mathfrak{H}$ as the compression of a minimal unitary dilation on $H$. Our model space $\mathfrak{H}$ is constructed in a canonical and conceptually more direct manner and doesn't depend on the Sz. Nagy-Foias characteristic function explicitly. We also show how a contraction can be constructed from a marked Nevanlinna disc, which is the geometric analogue of the characteristic function.