Subvarieties of the tetrablock and von Neumann's inequality

TL;DR

This paper explores the geometry of the tetrablock and its relation to operator theory, proving that distinguished varieties are one-dimensional and establishing von Neumann's inequality for certain operator triples.

Contribution

It characterizes distinguished varieties in the tetrablock via matrix representations and connects these to spectral sets and von Neumann's inequality for operator triples.

Findings

Every distinguished variety in the tetrablock is one-dimensional.

Von Neumann's inequality holds for operator triples with a specific subvariety.

Explicit dilation and functional models are constructed for these operator triples.

Abstract

We show an interplay between the complex geometry of the tetrablock $\mathbb E$ and the commuting triples of operators having $\overline{\mathbb E}$ as a spectral set. We prove that every distinguished variety in the tetrablock is one-dimensional and can be represented as \begin{equation}\label{eqn:1} \Omega=\{ (x_1,x_2,x_3)\in \mathbb E \,:\, (x_1,x_2) \in \sigma_T(A_1^*+x_3A_2\,,\, A_2^*+x_3A_1) \}, \end{equation} where $A_1,A_2$ are commuting square matrices of the same order satisfying $[A_1^*,A_1]=[A_2^*,A_2]$ and a norm condition. The converse also holds, i.e, a set of the form (\ref{eqn:1}) is always a distinguished variety in $\mathbb E$. We show that for a triple of commuting operators $\Upsilon = (T_1,T_2,T_3)$ having $\overline{\mathbb E}$ as a spectral set, there is a one-dimensional subvariety $\Omega_{\Upsilon}$ of $\overline{\mathbb E}$ depending on $\Upsilon$ such that von-Neumann's inequality holds, i.e, \[ f(T_1,T_2,T_3)\leq \sup_{(x_1,x_2,x_3)\in\Omega_{\Upsilon}}\, |f(x_1,x_2,x_3)|, \] for any holomorphic polynomial $f$ in three variables, provided that $T_3^n\rightarrow 0$ strongly as $n\rightarrow \infty$. The variety $\Omega_\Upsilon$ has been shown to have representation like (\ref{eqn:1}), where $A_1,A_2$ are the unique solutions of the operator equations \begin{gather*} T_1-T_2^*T_3=(I-T_3^*T_3)^{\frac{1}{2}}X_1(I-T_3^*T_3)^{\frac{1}{2}} \text{ and } \\ T_2-T_1^*T_3=(I-T_3^*T_3)^{\frac{1}{2}}X_2(I-T_3^*T_3)^{\frac{1}{2}}. \end{gather*} We also show that under certain condition, $\Omega_{\Upsilon}$ is a distinguished variety in $\mathbb E$. We produce an explicit dilation and a concrete functional model for such a triple $(T_1,T_2,T_3)$ in which the unique operators $A_1,A_2$ play the main role. Also, we describe a connection of this theory with the distinguished varieties in the bidisc and in the symmetrized bidisc.