Transference of Metrics Induced by Unitary Couplings, a Sarason Theorem for the Bidimensional Torus, and a Sz.-Nagy-Foias Theorem for Two Pairs of Dilations

Liftings of invariant forms defined in semi-invariant subspaces of the Hibert spaces of scattering systems with one or two evolution groups are obtained. These liftings provide direct proofs for the classical theorems of Sarason and of Sz.-Nagy and Foias, as well as a bidimensional version of Sarason′s theorem, and a theorem of Sz.-Nagy-Foias type for two pairs of dilations. The main tool used is a transference of a Hibert space metric induced by a coupling of isometries, of possible independent interest in the study of scattering systems.