Admissible Fundamental Operators and Models for $\Gamma_{E(3; 3; 1, 1, 1)}$-contraction and $\Gamma_{E(3; 2; 1, 2)}$-contraction

TL;DR

This paper develops models and characterizations for specific classes of multivariable contractions related to the domains (3; 3; 1, 1, 1) and (3; 2; 1, 2), including explicit constructions and dilation theories.

Contribution

It introduces new fundamental operator models and explicit constructions for (3; 3; 1, 1, 1) and (3; 2; 1, 2)-contractions, extending dilation and functional model theories.

Findings

Constructed (3; 3; 1, 1, 1)-unitaries from contractions.

Developed functional models for (3; 3; 1, 1, 1)- and (3; 2; 1, 2)-isometries.

Presented explicit dilation models including Douglas-type and Sz.-Nagy-Foias-type models.

Abstract

We show that for a given pure contraction $T_7$ acting on a Hilbert space $\mathcal{H}$, if $(\tilde{F}_1, \dots, \tilde{F}_6) \in \mathcal{B}(\mathcal{D}_{T^*_7})$ with $[\tilde{F}_i, \tilde{F}_j] = 0, [\tilde{F}^*_i, \tilde{F}_{7-j}] = [\tilde{F}^*_j, \tilde{F}_{7-i}]$,$w(\tilde{F}^*_i + \tilde{F}_{7-i}z) \leqslant 1$ and these operators satisfy \[(\tilde{F}^*_i + \tilde{F}_{7-i}z)\Theta_{T_7}(z) = \Theta_{T_7}(z)(F_i + F^*_{7-i}z) \,\, \text{for all} \,\, z \in \mathbb{D}\] for $1 \leqslant i, j \leqslant 6$ for some $(F_1, \dots, F_6) \in \mathcal{B}(\mathcal{D}_{T_7})$ with $w(F^*_i + F_{7-i}z) \leqslant 1$ for $1 \leqslant i \leqslant 6$, then there exists a $\Gamma_{E(3; 3; 1, 1, 1)}$-contraction $(T_1, \dots, T_7)$ such that $F_1, \dots, F_6$ are the fundamental operators of $(T_1, \dots, T_7)$ and $\tilde{F}_1, \dots, \tilde{F}_6$ are the fundamental operators of $(T^*_1, \dots, T^*_7)$. We also prove similar type of result for pure $\Gamma_{E(3; 2; 1, 2)}$-contraction. We explicitly construct a $\Gamma_{E(3; 3; 1, 1, 1)}$-unitary (respectively, a $\Gamma_{E(3; 2; 1, 2)}$-unitary) starting from a $\Gamma_{E(3; 3; 1, 1, 1)}$-contraction (respectively, a $\Gamma_{E(3; 2; 1, 2)}$-contraction). Further, we develop functional models for general $\Gamma_{E(3; 3; 1, 1, 1)}$-isometries (respectively, $\Gamma_{E(3; 2; 1, 2)}$-isometries). In particular, we construct Douglas-type and Sz.-Nazy-Foias-type models for $\Gamma_{E(3; 3; 1, 1, 1)}$-contractions (respectively, $\Gamma_{E(3; 2; 1, 2)}$-contractions). Finally, we present a Schaffer-type model for the $\Gamma_{E(3; 3; 1, 1, 1)}$-isometric dilation (respectively, the $\Gamma_{E(3; 2; 1, 2)}$-isometric dilation).