Functional Models for $Γ_{E(3; 3; 1, 1, 1)}$-contraction, $Γ_{E(3; 2; 1, 2)}$-contraction and Tetrablock contraction
Dinesh Kumar Keshari, Suryanarayan Nayak, Avijit Pal, Bhaskar Paul
TL;DR
This work develops a comprehensive operator-theoretic framework for $\Gamma_{E(3;3;1,1,1)}$- and $\Gamma_{E(3;2;1,2)}$-contractions, alongside tetrablock contractions, by defining fundamental operators, establishing fundamental equations, and constructing explicit Nagy–Foias-type functional representations on defect spaces. It proves complete unitary invariants for the pure cases via coincidence of the characteristic function and unitary equivalence of the fundamental operators, and extends the analysis to abstract dilation-type models for special c.n.u. classes under precise commutativity constraints. The results provide concrete model- and invariant-based tools for multivariable operator theory in these domains, with counterexamples highlighting the necessity of the imposed hypotheses. Together, these findings advance the understanding of spectral-set theory, operator tuple decompositions, and canonical representations in the tetrablock and related quotients.
Abstract
We obtain various characterizations of the fundamental operators of $Γ_{E(3; 3; 1, 1, 1)}$-contraction and $Γ_{E(3; 2; 1, 2)}$-contraction. We also demonstrate some important relations between the fundamental operators of a $Γ_{E(3; 3; 1, 1, 1)}$-contraction and a $Γ_{E(3; 2; 1, 2)}$-contraction. We describe functional models for \textit{pure $Γ_{E(3; 3; 1, 1, 1)}$-contraction} and \textit{pure $Γ_{E(3; 2; 1, 2)}$-contraction}. We give a complete set of unitary invariants for a pure $Γ_{E(3; 3; 1, 1, 1)}$-contraction and a pure $Γ_{E(3; 2; 1, 2)}$-contraction. We demonstrate the functional models for a certain class of completely non-unitary $Γ_{E(3; 3; 1, 1, 1)}$-contraction $\textbf{T} = (T_1, \dots, T_7)$ and completely non-unitary $Γ_{E(3; 2; 1, 2)}$-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, $Γ_{E(3; 3; 1, 1, 1)}$-contraction and $Γ_{E(3; 2; 1, 2)}$-contraction may not exist if we drop the hypothesis of the above equations, respectively..