Resumen de A model theory for operators associated with a domain related to μ-synthesis
Bappa Bisai, Sourav Pal
A commuting triple of Hilbert space operators (A, B, P) for which the closure of the tetrablock E, where E={(x1,x2,x3)∈C3:|x3|<1&x1=c1+x3c¯2,x2=c2+x3c¯1,c1,c2∈C with |c1|+|c2|<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 F1,F2∈B(DP), which are called the fundamental operators of (A, B, P), satisfying A−B∗P=DPF1DP,B−A∗P=DPF2DP.
An E-contraction (A, B, P) admits a canonical decomposition (A1⊕A2,B1⊕B2,P1⊕P2) into an E-unitary (A1,B1,P1) and a completely non-unitary (c.n.u.) E-contraction (A2,B2,P2). 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=C1+PC∗2,B=C2+PC∗1 for some operators C1,C2 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.
