Operator theory and representation of distinguished varieties in the symmetrized tridisc

Abstract

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} Λ=\{ (s_1,s_2,p)\in \mathbb G_3 \,:\, (s_1,s_2) \in σ_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 ${\mathbb{G}}_3$. 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, \[ f(S_1,S_2,P)\leq \sup_{(s_1,s_2,p)\inΛ_Σ}\, |f(s_1,s_2,p)|, \] for any holomorphic polynomial $f$ in three variables, provided that $P^n\to 0$ strongly as $n\to \infty$. 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 ${\mathbb{D}}^2$.

We withdraw this article because we could resolve the n-variable case

References in corpus (1)

• Extremal holomorphic maps and the symmetrised bidisc