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A model theory for Γ-contractions

Lookup NU author(s): Professor Jim Agler, Professor Nicholas Young


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A Γ-contraction is a pair of commuting operators on Hubert space for which the symmetrised bidisc Γ = def {(z1 + z2, z1z2 ) : |z1| ≤ 1, |z2| ≤ 1} ⊂ ℂ2 is a spectral set. We develop a model theory for such pairs which parallels a part of the well-known Nagy-Foiaş 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,Tz̄) where Tψ, Tz̄ are suitable block-Toeplitz 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.

Publication metadata

Author(s): Agler J, Young NJ

Publication type: Article

Publication status: Published

Journal: Journal of Operator Theory

Year: 2003

Volume: 49

Issue: 1

Pages: 45-60

Print publication date: 01/12/2003

ISSN (print): 0379-4024

ISSN (electronic):

Publisher: Academia Romana