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

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