On decomposition of operators having ${\Gamma }_3$ as a spectral set

Abstract

The symmetrized polydisc of dimension three is the set \[ Γ_3 =\{ (z_1+z_2+z_3, z_1z_2+z_2z_3+z_3z_1, z_1z_2z_3)\,:\, |z_i|\leq 1 \,,\, i=1,2,3 \} \subseteq \mathbb C^3\,. \] 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.

To appear in Operators and Matrices, 2017