Minimal isometric dilations and operator models for the polydisc

For commuting contractions $T_1,\dots,T_n$ acting on a Hilbert space $\mathscr{H}$ with $T=\prod_{i=1}^n T_i$ , we find a necessary and sufficient condition such that $(T_1,\dots,T_n)$ dilates to a commuting tuple of isometries $(V_1,\dots,V_n)$ on the minimal isometric dilation space of T with $V=\prod_{i=1}^nV_i$ being the minimal isometric dilation of T . This isometric dilation provides a commutant lifting of $(T_1, \dots, T_n)$ on the minimal isometric dilation space of T . We construct both Schäffer and Sz. Nagy–Foias-type isometric dilations for $(T_1,\dots,T_n)$ on the minimal dilation spaces of T . Also, a different dilation is constructed when the product T is a $C._0$ contraction, that is, ${T^*}^n \rightarrow 0$ as $n \rightarrow \infty$ . As a consequence of these dilation theorems, we obtain different functional models for $(T_1,\dots,T_n)$ 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.