A model theory for operators associated with a domain related to μ-synthesis.

A commuting triple of Hilbert space operators (A, B, P) for which the closure of the tetrablock E , where E = { (x 1 , x 2 , x 3) ∈ C 3 : | x 3 | < 1 & x 1 = c 1 + x 3 c ¯ 2 , x 2 = c 2 + x 3 c ¯ 1 , c 1 , c 2 ∈ C with | c 1 | + | c 2 | < 1 } , is a spectral set is called a tetrablock contraction or an E -contraction. To every E -contraction (A, B, P), there are unique operators F 1 , F 2 ∈ B (D P) , which are called the fundamental operators of (A, B, P), satisfying A - B ∗ P = D P F 1 D P , B - A ∗ P = D P F 2 D P. An E -contraction (A, B, P) admits a canonical decomposition (A 1 ⊕ A 2 , B 1 ⊕ B 2 , P 1 ⊕ P 2) into an E -unitary (A 1 , B 1 , P 1) and a completely non-unitary (c.n.u.) E -contraction (A 2 , B 2 , P 2) . As there already exists an easily understood model for an E -unitary in the literature, it suffices to restrict attention to the c.n.u. E -contraction part. Here we construct an explicit minimal E -isometric dilation for a c.n.u. E -contraction whose fundamental operators satisfy (1.1) below (a class for which it is known that such dilation exists). As a consequence of this explicit dilation, we obtain a functional model for the same class of E -contractions. With the help of this functional model we express an E -contraction (A, B, P) as A = C 1 + P C 2 ∗ , B = C 2 + P C 1 ∗ for some operators C 1 , C 2 and this representation is operator theoretic analogue of the representation of the points in E . We also construct a different functional model, which is not necessarily commutative, for a c.n.u. E -contraction (A, B, P) when A, B commute with P ∗ . This functional model is obtained even without having an E -isometric dilation exhibited in the model. We show by an example that such a model may not be possessed by (A, B, P) if the condition that A, B commute with P ∗ , is dropped from the hypothesis. A complete unitary invariant is achieved for a c.n.u. E -contraction (A, B, P) when A, B commute with P ∗ . The fundamental operators play the central role in all these constructions. Also, we produce a new characterization for an E -unitary. [ABSTRACT FROM AUTHOR]