Dilation and Functional Models for Pure $\mathbf{\Theta}_n$-Contractions and the von Neumann Inequality on Distinguished Varieties in $\mathbf{\Theta}_n$

Abstract

In this paper, we introduce the notion of a distinguished variety in the domain $\mathbf{\Theta}_n$. One of the main results of the paper is a determinantal representation for every distinguished variety in $\mathbf{\Theta}_n$. We also show that the closure of every distinguished variety is polynomially convex. Furthermore, we obtain a dilation and a functional model for a class of pure $\mathbf{\Theta}_n$-contractions. Finally, we show that for a $\mathbf{\Theta}_n$-contraction $\mathbf{T}=(T_1,\dots,T_n)$ such that $T_n^*$ is a pure contraction, there exists an algebraic variety in $\mathbf{\Theta}_n$ for which the von Neumann inequality holds on the intersection of the closure of the variety with the distinguished boundary of $\mathbf{\Theta}_n$.