On decomposition of operators having $\Gamma_3$ as a spectral set

TL;DR

This paper studies operators with the symmetrized 3-disc as a spectral set, proving they can be decomposed into a unitary part and a non-unitary part, and provides new characterizations of these sets and operators.

Contribution

It introduces a decomposition theorem for b3_3-contractions, paralleling classical contraction decompositions, and offers novel characterizations of b3_3 and related operators.

Findings

Every b3_3-contraction admits a decomposition into a b3_3-unitary and a non-unitary part.

New characterizations of the set b3_3 and b3_3-contractions.

The decomposition parallels the classical contraction decomposition into unitary and non-unitary parts.

Abstract

The symmetrized polydisc of dimension three is the set \[ \Gamma_3 =\{ (z_1+z_2+z_3, z_1z_2+z_2z_3+z_3z_1, z_1z_2z_3)\,:\, |z_i|\leq 1 \,,\, i=1,2,3 \} \subseteq \mathbb C^3\,. \] A triple of commuting operators for which $\Gamma_3$ is a spectral set is called a $\Gamma_3$-contraction. We show that every $\Gamma_3$-contraction admits a decomposition into a $\Gamma_3$-unitary and a completely non-unitary $\Gamma_3$-contraction. This decomposition parallels the canonical decomposition of a contraction into a unitary and a completely non-unitary contraction. We also find new characterizations for the set $\Gamma_3$ and $\Gamma_3$-contractions.