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
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