A model theory for $\Gamma$-contractions
Jim Agler and Nicholas Young
Keywords
functional model, operator theory
Status
Appeared in the Journal of Operator Theory
49(2003)45-60.
Abstract
A {\em $\Gamma$-contraction} is a pair of commuting
operators on Hilbert space for which the symmetrised
bidisc
$$
\Gamma\stackrel{=}{\rm def} \left\{(z_1+z_2, z_1z_2):
|z_1| \le 1, |z_2|\le 1\right\} \subset \mathbb{C}^2
$$
is a spectral set. We develop a model theory for such
pairs parallel to the well-known Nagy-Foias model for
contractions. In particular we show that any
$\Gamma$-contraction is unitarily equivalent to the
restriction to a joint invariant subspace of the
orthogonal direct sum of a $\Gamma$-unitary and a
``model $\Gamma$-contraction" of the form $(T_\psi,
T_{\overline z})$ where $T_\psi, T_{\overline z}$ are
suitable block-Toeplitz operators on a vectorial Hardy
space, and $\Gamma$-unitaries are defined to be pairs
of operators of the form $(U_1+U_2, U_1U_2)$ for some
pair $U_1, U_2$ of commuting unitaries.
Nicholas Young
26 July 2000.