A functional model for pure $\Gamma$-contractions

Abstract

A pair of commuting operators $(S,P)$ defined on a Hilbert space $\mathcal{H}$ for which the closed symmetrized bidisc $$\Gamma =\{(z_1+z_2,z_1{z}_2)::|z_1|\le 1,|z_2|\le 1\}\subseteq {\mathbb{C}}^2,$$ 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, ${P^{\ast }}^n\to 0$ strongly as $n\to \infty$. 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 $$S-S^{\ast }P=D_{P}X{D}_P,\text{where}X\in \mathcal{B}({\mathcal{D}}_P),$$and is called the fundamental operator of the $\Gamma$-contraction $(S,P)$. We also discuss some important properties of the fundamental operator.

Journal of Operator Theory, 71 (2014), 327-339