Functional Models for $\Gamma_{E(3; 3; 1, 1, 1)}$-contraction, $\Gamma_{E(3; 2; 1, 2)}$-contraction and Tetrablock contraction

TL;DR

This paper characterizes fundamental operators, develops functional models, and identifies unitary invariants for specific classes of contractions related to the tetrablock and certain domains, advancing the understanding of their structure and classification.

Contribution

It introduces new functional models and complete invariants for pure and non-unitary contractions associated with the domains mma_{E(3; 3; 1, 1, 1)} and mma_{E(3; 2; 1, 2)}, including relations between their fundamental operators.

Findings

Functional models for pure contractions are established.

Complete sets of unitary invariants are provided.

Counterexamples show the necessity of certain conditions.

Abstract

We obtain various characterizations of the fundamental operators of $\Gamma_{E(3; 3; 1, 1, 1)}$-contraction and $\Gamma_{E(3; 2; 1, 2)}$-contraction. We also demonstrate some important relations between the fundamental operators of a $\Gamma_{E(3; 3; 1, 1, 1)}$-contraction and a $\Gamma_{E(3; 2; 1, 2)}$-contraction. We describe functional models for \textit{pure $\Gamma_{E(3; 3; 1, 1, 1)}$-contraction} and \textit{pure $\Gamma_{E(3; 2; 1, 2)}$-contraction}. We give a complete set of unitary invariants for a pure $\Gamma_{E(3; 3; 1, 1, 1)}$-contraction and a pure $\Gamma_{E(3; 2; 1, 2)}$-contraction. We demonstrate the functional models for a certain class of completely non-unitary $\Gamma_{E(3; 3; 1, 1, 1)}$-contraction $\textbf{T} = (T_1, \dots, T_7)$ and completely non-unitary $\Gamma_{E(3; 2; 1, 2)}$-contraction $\textbf{S} = (S_1, S_2, S_3, \tilde{S}_1, \tilde{S}_2)$ which satisfy the following conditions: \begin{equation}\label{Condition 1} \begin{aligned} &T^*_iT_7 = T_7T^*_i \,\, \text{for} \,\, 1 \leqslant i \leqslant 6 \end{aligned} \end{equation} and \begin{equation}\label{Condition 2} \begin{aligned} &S^*_iS_3 = S_3S^*_i, \tilde{S}^*_jS_3 = S_3\tilde{S}^*_j \,\, \text{for} \,\, 1 \leqslant i, j \leqslant 2, \end{aligned} \end{equation} respectively. We also describe a functional model for a completely non-unitary tetrablock contraction $\textbf{T} = (A_1,A_2,P)$ that satisfies \begin{equation}\label{Condition 3} \begin{aligned} A^*_iP = PA^*_i \,\, \text{for $1 \leqslant i \leqslant 2$}. \end{aligned} \end{equation} By exhibiting counter examples, we show that such abstract model of tetrablock contraction, $\Gamma_{E(3; 3; 1, 1, 1)}$-contraction and $\Gamma_{E(3; 2; 1, 2)}$-contraction may not exist if we drop the hypothesis of the above equations, respectively..