Canonical decomposition of operators associated with the symmetrized polydisc

TL;DR

This paper establishes a canonical decomposition for operators associated with the symmetrized polydisc, analogous to classical contraction decompositions, and introduces new characterizations of these operator sets.

Contribution

It proves that every b3_n-contraction can be uniquely decomposed into a b3_n-unitary and a non-unitary part, extending classical operator theory.

Findings

Decomposition of b3_n-contractions into b3_n-unitary and non-unitary parts.

New characterizations of b3_n and b3_n-contractions.

Analogue of the canonical contraction decomposition for symmetrized polydisc operators.

Abstract

A tuple of commuting operators $(S_1,\dots,S_{n-1},P)$ for which the closed symmetrized polydisc $\Gamma_n$ is a spectral set is called a $\Gamma_n$-contraction. We show that every $\Gamma_n$-contraction admits a decomposition into a $\Gamma_n$-unitary and a completely non-unitary $\Gamma_n$-contraction. This decomposition is an analogue to the canonical decomposition of a contraction into a unitary and a completely non-unitary contraction. We also find new characterizations for the set $\Gamma_n$ and $\Gamma_n$-contractions.