A functional model for pure Γ-contractions

Abstract

A pair of commuting operators (S,P) defined on a Hilbert space H for which the closed symmetrized bidisc Γ = {(z₁+z₂, z₁z₂):|z₁| ≤ 1,|z₂| ≤ 1} ⊆ C² is a spectral set is called a Γ-contraction in the literature. A Γ-contraction (S,P) is said to be pure if P is a pure contraction, i.e., P*ⁿ → 0 strongly as n → ∞. Here we construct a functional model and produce a set of unitary invariants for a pure Γ-contraction. The key ingredient in these constructions is an operator, which is the unique solution of the operator equation S-S*P=D_pXD_p, where X ϵ B(D_p),and is called the fundamental operator of the Γ-contraction (S,P). We also discuss some important properties of the fundamental operator.