Part I contains a revised version of the coordinate-free function model of a Hubert space contraction, based on analysis of functional embeddings related to the minimal unitary dilation of the operator. Using functional embeddings, we introduce and study all other objects of model theory, including the characteristic function, various concrete forms (transcriptions) of the model, one-sided resolvents, and so on. For the case of an inner scalar characteristic function we develop the classical H∞-calculus up to a local function calculus on the level curves of the characteristic function. The spectrum of operators commuting with the model operator, and in particular functions of the latter, are described in terms of their liftings. A simplified proof of the invariant subspace theorem is given, using the functional embeddings and regular factorizations of the characteristic function. As examples, we consider some compact convolution-type integral operators, and dissipative Schrödinger (Sturm-Liouville) operators on the half-line.

Part II, which will appear elsewhere, will contain applications of the model approach to such topics as angles between invariant subspaces and operator corona equations, generalized spectral decompositions and free interpolation problems, resolvent criteria for similarity to a normal operator, and weak generators of the commutant and the reflexivity property. Classical topics of stability of the continuous spectrum and scattering theory will also be brought into the fold of the coordinate-free model approach.