Journal of Operator Theory

Volume 49, Issue 1, Winter 2003 pp. 45-60.

A model theory for

$\Gamma$ -contractions

Authors: J. Agler

$1$ and N.J. Young

$2$
Author institution:

$1$ Department of Mathematics, University of California at San Diego, La Jolla, CA 92093, USA

$2$ Department of Mathematics, University of Newcastle upon Tyne, Merz Court, Newcastle upon Tyne NE1 7RU, UK

Summary: 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 which parallels a part of the well-known Nagy-Foiași 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.

Keywords: model operator, spectral set, symmetrised bidisc

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