Admissible Fundamental Operators and Models for $Γ_{E(3; 3; 1, 1, 1)}$-contraction and $Γ_{E(3; 2; 1, 2)}$-contraction
Avijit Pal, Bhaskar Paul
TL;DR
The paper develops a comprehensive dilation and modeling framework for Γ_E(3;3;1,1,1)-contractions and Γ_E(3;2;1,2)-contractions by identifying admissible fundamental operators and establishing canonical unitary extensions. It then provides interchangeable functional models—Douglas-type and Sz.-Nagy–Foias-type—for completely non-unitary and pure cases, and constructs Schäffer-type isometric dilations to realize these tuples as concrete operator models on Hardy-type spaces. The results yield explicit canonical representations, unique liftings to unitary dilations, and model-space descriptions that unify dilation theory with operator-model theory in several complex variables. Collectively, these contributions advance the understanding of multivariable operator tuples associated with these tetrablock-like domains and provide practical tools for analyzing their spectral and dilation properties.
Abstract
We show that for a given pure contraction $T_7$ acting on a Hilbert space $\mathcal{H}$, if $(\tilde{F}_1, \dots, \tilde{F}_6) \in \mathcal{B}(\mathcal{D}_{T^*_7})$ 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 \[(\tilde{F}^*_i + \tilde{F}_{7-i}z)Θ_{T_7}(z) = Θ_{T_7}(z)(F_i + F^*_{7-i}z) \,\, \text{for all} \,\, z \in \mathbb{D}\] 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 $Γ_{E(3; 3; 1, 1, 1)}$-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 $Γ_{E(3; 2; 1, 2)}$-contraction. We explicitly construct a $Γ_{E(3; 3; 1, 1, 1)}$-unitary (respectively, a $Γ_{E(3; 2; 1, 2)}$-unitary) starting from a $Γ_{E(3; 3; 1, 1, 1)}$-contraction (respectively, a $Γ_{E(3; 2; 1, 2)}$-contraction). Further, we develop functional models for general $Γ_{E(3; 3; 1, 1, 1)}$-isometries (respectively, $Γ_{E(3; 2; 1, 2)}$-isometries). In particular, we construct Douglas-type and Sz.-Nazy-Foias-type models for $Γ_{E(3; 3; 1, 1, 1)}$-contractions (respectively, $Γ_{E(3; 2; 1, 2)}$-contractions). Finally, we present a Schaffer-type model for the $Γ_{E(3; 3; 1, 1, 1)}$-isometric dilation (respectively, the $Γ_{E(3; 2; 1, 2)}$-isometric dilation).