Minimal isometric dilations and operator models for the polydisc

For commuting contractions T1, . . ., Tn acting on a Hilbert space H with T =^Qn_i=1 Ti, we find a necessary and sufficient condition such that (T1, . . ., Tn) dilates to a commuting tuple of isometries (V1, . . ., Vn) on the minimal isometric dilation space of T with V =^Qn_i=1 Vi being the minimal isometric dilation of T. This isometric dilation provides a commutant lifting of (T1, . . ., Tn) on the minimal isometric dilation space of T. We construct both Schäffer and Sz. Nagy–Foias-type isometric dilations for (T1, . . ., Tn) on the minimal dilation spaces of T. Also, a different dilation is constructed when the product T is a C.0 contraction, that is, T^∗n → 0 as n → ∞. As a consequence of these dilation theorems, we obtain different functional models for (T1, . . ., Tn) in terms of multiplication operators on vectorial Hardy spaces. One notable fact about our models is that the multipliers are all analytic functions in one variable. The dilation when T is a C.0 contraction leads to a conditional factorization of T. Several examples have been constructed.