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 $[F_1^{\ast },F_1]=[F_2^{\ast },F_2]$ and a norm condition. The converse also holds, i.e, a set of the form (\ref{eqn:1}) is always a distinguished variety in $\mathbb G_3$\mathrm{.}Weshowthatforatripleofcommutingoperators$\Sigma = (S_1,S_2,P)$having$\Gamma_3$asaspectralset,thereisaone-dimensionalsubvariety$\Lambda_{\Sigma}$of$\Gamma_3$dependingon$\Sigma$ such that von-Neumann’s inequality holds, i.e, $$f(S_1,S_2,P)\le \underset{(s_1,s_2,p)\in {\mathrm{\Lambda }}_{\mathrm{\Sigma }}}{\sup }\mathrm{\mid }f(s_1,s_2,p)\mathrm{\mid },$$ for any holomorphic polynomial $f$ in three variables, provided that $P^n\to 0$ strongly as $n\to \mathrm{\infty }$. The variety ${\mathrm{\Lambda }}_{\mathrm{\Sigma }}$ 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, ${\mathrm{\Lambda }}_{\mathrm{\Sigma }}$ 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 ${\mathbb{D}}^2$.