# Journal of Operator Theory

Volume 71, Issue 2, Spring 2014  pp. 327-339.

A functional model for pure $\Gamma$-contractions

Authors:  Tirthankar Bhattacharyya (1) and Sourav Pal (2)
Author institution: (1) Department of Mathematics, Indian Institute of Science, Bangalore, 560012, India
(2) Department of Mathematics, Indian Institute of Science, Bangalore, 560012, India

Summary:  A pair of commuting operators $(S,P)$ defined on a Hilbert space $\mathcal H$ for which the closed symmetrized bidisc $\Gamma= \{ (z_1+z_2,z_1z_2): |z_1|\leqslant 1,\, |z_2|\leqslant 1 \} \subseteq \mathbb C^2$ 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 set of unitary invariants 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.

DOI: http://dx.doi.org/10.7900/jot.2012mar21.1946
Keywords: symmetrized bidisc, fundamental operator, functional model, unitary invariants