A functional model for pure $\Gamma$-contractions

A pair of commuting operators $(S,P)$ defined on a Hilbert space $\mathcal H$ for which the closed symmetrized bidisc $$\Gamma= \{(z_1+z_2,z_1z_2):: |z_1|\leq 1,\, |z_2|\leq 1 \}\subseteq \mathbb C^2,$$ is a spectral set is called a $\Gamma$-contraction in the literature. A $\Gamma$-contraction $(S,P)$ is said to be pure if $P$ is a pure contraction, i.e, ${P^*}^n \rightarrow 0$ strongly as $n \rightarrow \infty$. Here we construct a functional model and produce a complete unitary invariant for a pure $\Gamma$-contraction. The key ingredient in these constructions is an operator, which is the unique solution of the operator equation $$S-S^*P=D_PXD_P, \textup{where} X\in \mathcal B(\mathcal D_P),$$and is called the fundamental operator of the $\Gamma$-contraction $(S,P)$. We also discuss some important properties of the fundamental operator.