Two classical theorems of function model theory via the coordinate-free approach

The aim of the paper is to present a new approach to the proof of two well-known theorems of Sz.-Hagy–Foiaş: the first one concerns the correspondence between invariant subspaces of a given contraction $T$ and regular factorizations of the characteristic function $\theta_T$ of $T$, the second one is the commutant lifting theorem. The proofs are based on the coordinate-free approach to the functional model. In other words, a concrete spectral representation of a minimal unitary dilation is not fixed. The essential point in the first theorem is an assertion in terms of functional mappings $\eta: L^2(F)\longmapsto\mathcal{H}$ ($\mathcal{H}$ is the space of a minimal unitary dilation $U$) equivalent to the existence of an invariant supspace of $T$. As to the lifting theorem, our approach provides us with a new parametrization of lifted operator that seems to be more natural than the known Sz.-Nagy–Foiaş parametrization.