Dilation theory and functional models for tetrablock contractions

Abstract

A classical result of Sz.-Nagy asserts that a Hilbert space contraction operator $T$ can be dilated to a unitary $\cU$. A more general multivariable setting for these ideas is the setup where (i) the unit disk is replaced by a domain $\Omega$ contained in ${\mathbb C}^d$, (ii) the contraction operator $T$ is replaced by a commuting tuple $\bfT = (T_1, \dots, T_d)$ such that $\| r(T_1, \dots, T_d) \|_{\cL(\cH)} \le \sup_{\lam \in \Omega} | r(\lam) |$ for all rational functions with no singularities in $\overline{\Omega}$ and the unitary operator $\cU$ is replaced by an $\Omega$-unitary operator tuple, i.e., a commutative operator $d$-tuple $\bfU = (U_1, \dots, U_d)$ of commuting normal operators with joint spectrum contained in the distinguished boundary $b\Omega$ of $\Omega$. For a given domain $\Omega \subset {\mathbb C}^d$, the {\em rational dilation question} asks: given an $\Omega$-contraction $\bfT$ on $\cH$, is it always possible to find an $\Omega$-unitary $\bfU$ on a larger Hilbert space $\cK \supset \cH$ so that, for any $d$-variable rational function without singularities in $\overline{\Omega}$, one can recover $r(T)$ as $r(T) = P_\cH r(\bfU)|_\cH$. We focus here on the case where $\Omega $ is the {\em tetrablock}. (i) We identify a complete set of unitary invariants for a ${\mathbb E}$-contraction $(A,B,T)$ which can then be used to write down a functional model for $(A,B,T)$, thereby extending earlier results only done for a special case, (ii) we identify the class of {\em pseudo-commutative ${\mathbb E}$-isometries} (a priori slightly larger than the class of ${\mathbb E}$-isometries) to which any ${\mathbb E}$-contraction can be lifted, and (iii) we use our functional model to recover an earlier result on the existence and uniqueness of a ${\mathbb E}$-isometric lift $(V_1, V_2, V_3)$ of a special type for a ${\mathbb E}$-contraction $(A,B,T)$.