Operator theory and representation of distinguished varieties in the symmetrized tridisc

We show that every distinguished variety in the symmetrized tridisc $\mathbb G_3$ is one-dimensional and can be represented as \begin{equation}\label{eqn:1} \Lambda=\{ (s_1,s_2,p)\in \mathbb G_3 \,:\, (s_1,s_2) \in \sigma_T(F_1^*+pF_2\,,\, F_2^*+pF_1) \}, \end{equation} where $F_1,F_2$ are commuting square matrices of the same order satisfying $[F_1^*,F_1]=[F_2^*,F_2]$ and a norm condition. The converse also holds, i.e, a set of the form (\ref{eqn:1}) is always a distinguished variety in $\mathbb G_3$. We show that for a triple of commuting operators $\Sigma = (S_1,S_2,P)$ having $\Gamma_3$ as a spectral set, there is a one-dimensional subvariety $\Lambda_{\Sigma}$ of $\Gamma_3$ depending on $\Sigma$ such that von-Neumann's inequality holds, i.e, $$f(S_1,S_2,P)\leq \sup_{(s_1,s_2,p)\in\Lambda_{\Sigma}}\, |f(s_1,s_2,p)|,$$ for any holomorphic polynomial $f$ in three variables, provided that $P^n\rightarrow 0$ strongly as $n\rightarrow \infty$. The variety $\Lambda_\Sigma$ has been shown to have representation like (\ref{eqn:1}), where $F_1,F_2$ are the unique solutions of the operator equations \begin{gather*} S_1-S_2^*P=(I-P^*P)^{\frac{1}{2}}X_1(I-P^*P)^{\frac{1}{2}} \text{ and } \\ S_2-S_1^*P=(I-P^*P)^{\frac{1}{2}}X_2(I-P^*P)^{\frac{1}{2}}. \end{gather*} We also show that under certain condition, $\Lambda_\Sigma$ is a distinguished variety in $\mathbb G_3$. We produce an explicit dilation and a concrete functional model for such a triple $(S_1,S_2,P)$ in which the unique operators $F_1,F_2$ play the main role. Also, we describe a connection of this theory with the distinguished varieties in the symmetrized bidisc and in the unit bidisc $\mathbb D^2$.