A model theory for Γ-contractions

A Γ-contraction is a pair of commuting operators on Hilbert space for which the symmetrised bidisc Γ def = {(z1 + z2, z1z2) : |z1| 6 1, |z2| 6 1} ⊂ C 2 is a spectral set. We develop a model theory for such pairs which parallels a part of the well-known Nagy-Foias model for contractions. In particular we show that any Γ-contraction is unitarily equivalent to the restriction to a joint invariant subspace of the orthogonal direct sum of a Γ-unitary and a “model Γ-contraction” of the form (Tψ, Tz) where Tψ, Tz are suitable blockToeplitz operators on a vectorial Hardy space, and Γ-unitaries are defined to be pairs of operators of the form (U1 + U2, U1U2) for some pair U1, U2 of commuting unitaries.