Almost isometric operators: a function model, invariant subspaces, the commutant

A new function model for an arbitrary bounded operator on Hilbert space is constructed. This model generalizes the model of Sz.-Nagy and Foiaş, for contractions and seems to be useful for operators close to an isometry (in a sense). All the model spaces are Hilbert spaces, but instead of dilation a generalization of it is used. The model admits a simmetry relative to the map $z\mapsto1/z$ of the complex plane. In terms of the model the question of lifting of the commutant is investigated, a relationship between invariant subspaces of a unitary operator is established, the characteristic function of the model operator is calculated. Some other problems are solved as well.