Functional Models for -contraction, -contraction and Tetrablock contraction
arXiv:2511.01644
Abstract
We obtain various characterizations of the fundamental operators of -contraction and -contraction. We also demonstrate some important relations between the fundamental operators of a -contraction and a -contraction. We describe functional models for \textit{pure -contraction} and \textit{pure -contraction}. We give a complete set of unitary invariants for a pure -contraction and a pure -contraction. We demonstrate the functional models for a certain class of completely non-unitary -contraction and completely non-unitary -contraction 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 that satisfies \begin{equation}\label{Condition 3} \begin{aligned} A^*_iP = PA^*_i \,\, \text{for }. \end{aligned} \end{equation} By exhibiting counter examples, we show that such abstract model of tetrablock contraction, -contraction and -contraction may not exist if we drop the hypothesis of the above equations, respectively..
This is the initial version of the paper. We will submit final version soon