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 $T_n^{\ast }$ 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 ${\mathbf{\Theta }}_n$.

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