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 as E = { ( x 1 , x 2 , x 3 ) ∈ C 3 : 1 − z x 1 − w x 2 + z w x 3 ≠ 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. [ABSTRACT FROM AUTHOR]