# 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-Foia\c{s} 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.

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