Dilation, functional model and a complete unitary invariant for $C._{0}\,\; \Gamma_n$-contractions

Abstract

A commuting tuple of operators $(S_1,\dots, S_{n-1},P)$, defined on a Hilbert space $\mathcal H$, for which the closed symmetrized polydisc \[ \Gamma_n =\left\{ \left(\sum_{1\leq i\leq n} z_i,\sum_{1\leq i<j\leq n}z_iz_j,\dots, \prod_{i=1}^n z_i \right): \,|z_i|\leq 1, i=1,\dots,n \right \} \] is a spectral set, is called a $\Gamma_n$-\textit{contraction}. A $\Gamma_n$-contraction is said to be \textit{pure} or $C._0$ if $P$ is $C._0$, that is, if ${P^*}^n \rightarrow 0$ strongly as $n \rightarrow \infty$. We show that for any $\Gamma_n$-contraction $(S_1,\dots, S_{n-1},P)$, there is a unique operator tuple $(A_1,\dots , A_{n-1})$ that satisfies the operator identities \[ S_i-S_{n-i}^*P=D_PA_iD_P\,, \quad \quad i=1,\dots, n-1. \] This unique tuple is called the \textit{fundamental operator tuple} or $\mathcal F_O$-tuple of $(S_1,\dots, S_{n-1},P)$. With the help of the $\mathcal F_O$-tuple, we construct an operator model for a $C._0 \; \Gamma_n$-contraction and show that there exist $n-1$ operators $C_1,\dots, C_{n-1}$ such that each $S_i$ can be represented as $S_i=C_i+PC_{n-i}^*$. We find an explicit minimal dilation for a class of $C._0 \; \Gamma_n$-contractions whose $\mathcal F_O$-tuples satisfy a certain condition. Also we establish that the $\mathcal F_O$-tuple of $(S_1^*,\dots, S_{n-1}^*,P^*)$ together with the characteristic function of $P$ constitute a complete unitary invariant for the $C._0$ $\Gamma_n$-contractions. The entire program is an analogue of the Nagy-Foias theory for $C._0$ contractions.