Minimal isometric dilations and operator models for the polydisc

TL;DR

This paper establishes conditions for commuting contractions to dilate to commuting isometries, constructs various dilation models, and derives functional models using analytic multipliers on Hardy spaces, advancing operator theory on the polydisc.

Contribution

It provides necessary and sufficient conditions for dilations of commuting contractions to commuting isometries and constructs new dilation models with analytic multipliers.

Findings

Characterization of dilation conditions for commuting contractions.

Construction of Sch"affer and Sz.-Nagy-Foias type dilations.

Functional models using analytic multipliers on Hardy spaces.

Abstract

For commuting contractions $T_1,\dots ,T_n$ acting on a Hilbert space $\mathcal H$ with $T=\prod_{i=1}^n T_i$, we find a necessary and sufficient condition under which $(T_1,\dots ,T_n)$ dilates to commuting isometries $(V_1,\dots ,V_n)$ on the minimal isometric dilation space $T$, where $V=\prod_{i=1}^nV_i$ is the minimal isometric dilation of $T$. We construct both Sch$\ddot{a}$ffer 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 analytic functions in one variable. The dilation, when $T$ is a $C._0$ contraction, leads to a conditional factorization of a $T$. Several examples have been constructed.