[Paper Review] The failure of rational dilation on the symmetrized $n$-disk for any $n\geq 3$

Sourav Pal|arXiv (Cornell University)|Dec 14, 2017

Holomorphic and Operator Theory27 references3 citations

TL;DR

This paper demonstrates that rational dilation fails on the symmetrized $n$-disc for all $n \geq 3$ by constructing a counterexample. It introduces new characterizations of points in $\mathbb{G}_n$ and $\Gamma_n$, and provides novel characterizations of $\Gamma_n$-unitaries and $\Gamma_n$-isometries, advancing the understanding of operator theory on symmetric domains.

ABSTRACT

The open and closed extit{symmetrized polydisc} or, extit{symmetrized $n$-disc} for $n\geq 2$, are the following subsets of $\mathbb C^n$: \begin{align*} \mathbb G_n &=\left\{ \left(\sum_{1\leq i\leq n} z_i,\sum_{1\leq i<j\leq n}z_iz_j,\dots, \prod_{i=1}^n z_i ight): \,|z_i|< 1, i=1,\dots,n ight \}, \Gamma_n & =\left\{ \left(\sum_{1\leq i\leq n} z_i,\sum_{1\leq i<j\leq n}z_iz_j,\dots, \prod_{i=1}^n z_i ight): \,|z_i|\leq 1, i=1,\dots,n ight \}. \end{align*} A tuple of commuting $n$ operators $(S_1,\dots,S_{n-1},P)$ defined on a Hilbert space $\mathcal H$ for which $\Gamma_n$ is a spectral set is called a $\Gamma_n$-contraction. In this article, we show by a counter example that rational dilation fails on the symmetrized $n$-disc for any $n\geq 3$. We find new characterizations for the points in $\mathbb G_n$ and $\Gamma_n$. We also present few new characterizations for the $\Gamma_n$-unitaries and $\Gamma_n$-isometries.

Motivation & Objective

• To investigate whether rational dilation holds on the symmetrized $n$-disc for $n \geq 3$, a key question in multivariable operator theory.
• To resolve the failure of rational dilation in this setting by constructing a counterexample.
• To provide new characterizations of points in the open and closed symmetrized polydiscs, $\mathbb{G}_n$ and $\Gamma_n$, respectively.
• To establish new structural characterizations of $\Gamma_n$-unitaries and $\Gamma_n$-isometries.
• To deepen the understanding of spectral sets and operator models in symmetric domains of $\mathbb{C}^n$.

Proposed method

• Construction of a specific $\Gamma_n$-contraction that does not admit a rational dilation, serving as a counterexample to the rational dilation property.
• Use of symmetric polynomial mappings to define the symmetrized $n$-disc as the image of the polydisc under the elementary symmetric functions.
• Analysis of the spectral properties of operators in the $\Gamma_n$-contraction framework using functional calculus on symmetric domains.
• Derivation of necessary and sufficient conditions for a point to lie in $\mathbb{G}_n$ and $\Gamma_n$ via polynomial and algebraic constraints on the variables.
• Application of operator-theoretic techniques to characterize $\Gamma_n$-unitaries and $\Gamma_n$-isometries through their functional models.
• Leveraging the structure of the symmetrized polydisc to reveal obstructions to rational dilation in higher dimensions.

Experimental results

Research questions

• RQ1 Does rational dilation hold for $\Gamma_n$-contractions on the symmetrized $n$-disc when $n \geq 3$?
• RQ2 What are the precise algebraic and analytic conditions that characterize points in $\mathbb{G}_n$ and $\Gamma_n$?
• RQ3 How can $\Gamma_n$-unitaries and $\Gamma_n$-isometries be characterized beyond their standard definitions?
• RQ4 What structural obstructions prevent rational dilation in the symmetrized $n$-disc for $n \geq 3$?
• RQ5 Can a counterexample be explicitly constructed to disprove rational dilation in this setting?

Key findings

• Rational dilation fails on the symmetrized $n$-disc for all $n \geq 3$, as demonstrated by a concrete counterexample of a $\Gamma_n$-contraction without a rational dilation.
• New necessary and sufficient conditions are derived for a point to belong to $\mathbb{G}_n$ and $\Gamma_n$, based on symmetric polynomial invariants and modulus constraints on the variables.
• The paper provides a complete characterization of $\Gamma_n$-unitaries as those tuples for which the joint spectrum lies in the distinguished boundary of $\Gamma_n$.
• It establishes that $\Gamma_n$-isometries are precisely those $\Gamma_n$-contractions for which the joint spectrum lies in the closure of the symmetrized polydisc and satisfies a specific functional model condition.
• The results reveal that the failure of rational dilation is intrinsic to the geometry of $\Gamma_n$ for $n \geq 3$, distinguishing it from the $n=2$ case.
• The new characterizations of $\Gamma_n$-unitaries and $\Gamma_n$-isometries provide a refined framework for studying operator models on symmetric domains.