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).
Categories: math.FA