A model theory for operators associated with a domain related to μ-synthesis
-
Bisai, Bappa [1] ; Pal, Sourav [1]
-
[1] Mathematics Department, Indian Institute of Technology Bombay, Powai, Mumbai, 400076, India
-
- Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 74, Fasc. 1, 2023, págs. 173-198
- Idioma: inglés
- DOI: 10.1007/s13348-021-00341-6
- Texto completo no disponible (Saber más ...)
-
Resumen
A commuting triple of Hilbert space operators (A, B, P) for which the closure of the tetrablock {\mathbb {E}}, where \begin{aligned} {\mathbb {E}} = \{ (x_1,x_2,x_3)\in {\mathbb {C}}^3 : \; |x_3|<1 \; \& \; x_1=c_1+ x_3\bar{c}_2\,, x_2=c_2 +x_3\bar{c}_1 \,, \; c_1,c_2 \in {\mathbb {C}} \text { with } |c_1|+|c_2|<1 \}, \end{aligned} is a spectral set is called a tetrablock contraction or an {\mathbb {E}}-contraction. To every {\mathbb {E}}-contraction (A, B, P), there are unique operators F_1,F_2 \in \mathcal B(\mathcal D_P), which are called the fundamental operators of (A, B, P), satisfying \begin{aligned} A-B^*P=D_PF_1D_P\;,\; B-A^*P=D_PF_2D_P. \end{aligned} An {\mathbb {E}}-contraction (A, B, P) admits a canonical decomposition (A_1\oplus A_2, B_1 \oplus B_2, P_1 \oplus P_2) into an {\mathbb {E}}-unitary (A_1,B_1,P_1) and a completely non-unitary (c.n.u.) {\mathbb {E}}-contraction (A_2,B_2,P_2). As there already exists an easily understood model for an {\mathbb {E}}-unitary in the literature, it suffices to restrict attention to the c.n.u. {\mathbb {E}}-contraction part. Here we construct an explicit minimal {\mathbb {E}}-isometric dilation for a c.n.u. {\mathbb {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 {\mathbb {E}}-contractions. With the help of this functional model we express an {\mathbb {E}}-contraction (A, B, P) as \begin{aligned} A=C_1+PC_2^*\;,\; B= C_2+PC_1^* \end{aligned} for some operators C_1,C_2 and this representation is operator theoretic analogue of the representation of the points in {\mathbb {E}}. We also construct a different functional model, which is not necessarily commutative, for a c.n.u. {\mathbb {E}}-contraction (A, B, P) when A, B commute with P^*. This functional model is obtained even without having an {\mathbb {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. {\mathbb {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 {\mathbb {E}}-unitary.
