# 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

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