Canonical decomposition of a tetrablock contraction and operator model

A triple of commuting operators for which the closed tetrablock E‾ is a spectral set is called a tetrablock contraction or an E-contraction. The set E is defined asE={(x1,x2,x3)∈C3:1−zx1−wx2+zwx3≠0 whenever |z|≤1,|w|≤1}. We show that every E-contraction can be uniquely written as a direct sum of an E-unitary and a completely non-unitary 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.