Canonical decomposition of a tetrablock contraction and operator model

TL;DR

This paper establishes a canonical decomposition for tetrablock contractions, showing they can be uniquely expressed as a sum of an E-unitary and a non-unitary part, and provides a concrete operator model for these triples.

Contribution

It introduces a unique decomposition for E-contractions analogous to classical contraction theory and constructs a specific operator model under certain conditions.

Findings

Every E-contraction can be uniquely decomposed into an E-unitary and a non-unitary part.

The decomposition parallels the classical contraction operator decomposition.

A concrete operator model for E-contractions satisfying specific conditions is developed.

Abstract

A triple of commuting operators for which the closed tetrablock $\overline{\mathbb E}$ is a spectral set is called a tetrablock contraction or an $\mathbb E$-contraction. The set $\mathbb E$ is defined as \[ \mathbb E = \{ (x_1,x_2,x_3)\in\mathbb C^3\,:\, 1-zx_1-wx_2+zwx_3\neq 0 \textup{ whenever } |z|\leq 1, |w|\leq 1 \}. \] We show that every $\mathbb E$-contraction can be uniquely written as a direct sum of an $\mathbb E$-unitary and a completely non-unitary $\mathbb E$-contraction. It is analogous to the canonical decomposition of a contraction operator into a unitary and a completely non-unitary contraction. We produce a concrete operator model for such a triple satisfying some conditions.