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

We obtain various characterizations of the fundamental operators of ${\mathrm{\Gamma }}_{E(3;3;1,1,1)}$-contraction and ${\mathrm{\Gamma }}_{E(3;2;1,2)}$-contraction. We also demonstrate some important relations between the fundamental operators of a ${\mathrm{\Gamma }}_{E(3;3;1,1,1)}$-contraction and a ${\mathrm{\Gamma }}_{E(3;2;1,2)}$-contraction. We describe functional models for $\mathit{\text{pure}}{\mathrm{\Gamma }}_{E(3;3;1,1,1)}\mathit{\text{-contraction}}$ and $\mathit{\text{pure}}{\mathrm{\Gamma }}_{E(3;2;1,2)}\mathit{\text{-contraction}}$. We give a complete set of unitary invariants for a pure ${\mathrm{\Gamma }}_{E(3;3;1,1,1)}$-contraction and a pure ${\mathrm{\Gamma }}_{E(3;2;1,2)}$-contraction. We demonstrate the functional models for a certain class of completely non-unitary ${\mathrm{\Gamma }}_{E(3;3;1,1,1)}$-contraction $\mathbf{\text{T}}=(T_1,\dots ,T_7)$ and completely non-unitary ${\mathrm{\Gamma }}_{E(3;2;1,2)}$-contraction $\mathbf{\text{S}}=(S_1,S_2,S_3,{\tilde{S}}_1,{\tilde{S}}_2)$ 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 $\mathbf{\text{T}}=(A_1,A_2,P)$ that satisfies \begin{equation}\label{Condition 3} \begin{aligned} A^_iP = PA^_i ,, $\text{for }1⩽i⩽2$. \end{aligned} \end{equation} By exhibiting counter examples, we show that such abstract model of tetrablock contraction, ${\mathrm{\Gamma }}_{E(3;3;1,1,1)}$-contraction and ${\mathrm{\Gamma }}_{E(3;2;1,2)}$-contraction may not exist if we drop the hypothesis of the above equations, respectively…