Canonical decomposition of a tetrablock contraction and operator model

A triple of commuting operators for which the closed tetrablock $\bar{\mathbb{E}}$ is a spectral set is called a tetrablock contraction or an $\mathbb{E}$-contraction. The set $\mathbb{E}$ is defined as $$\mathbb{E}=\{(x_1,x_2,x_3)\in {\mathbb{C}}^3:1-z{x}_1-w{x}_2+zw{x}_3\mathrm{\ne }0\text{ whenever }\mathrm{\mid }z\mathrm{\mid }\le 1,\mathrm{\mid }w\mathrm{\mid }\le 1\}\mathrm{.}$$ 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.