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

Abstract

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 \[(\tilde{F}^*_i + \tilde{F}_{7-i}z)Θ_{T_7}(z) = Θ_{T_7}(z)(F_i + F^*_{7-i}z) \,\, \text{for all} \,\, z \in \mathbb{D}\] 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 ${\Gamma }_{E(3;3;1,1,1)}$-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 ${\Gamma }_{E(3;2;1,2)}$-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).

This is the initial version of the paper. Final version will submit soon