The symmetrization map and Γ -contractions

The symmetrization map π: C²→ C² is defined by π(z₁, z₂) = (z₁+ z₂, z₁z₂). The closed symmetrized bidisc Γ is the symmetrization of the closed unit bidisc D²¯ , that is, Γ=π(D2¯)={(z1+z2,z1z2):|zi|≤1,i=1,2}. A pair of commuting Hilbert space operators (S, P) for which Γ is a spectral set is called a Γ -contraction. Unlike the scalars in Γ , a Γ -contraction may not arise as a symmetrization of a pair of commuting contractions, even not as a symmetrization of a pair of commuting bounded operators. We characterize all Γ -contractions which are symmetrization of pairs of commuting contractions. We show by constructing a family of examples that even if a Γ -contraction (S, P) = (T₁+ T₂, T₁T₂) for a pair of commuting bounded operators T₁, T₂ , no real number less than 2 can be a bound for the set { ‖ T₁‖ , ‖ T₂‖ } in general. Then we prove that every Γ -contraction (S, P) is the restriction of a Γ -contraction (S~ , P~) to a common reducing subspace of S~ , P~ and that (S~ , P~) = (A₁+ A₂, A₁A₂) for a pair of commuting operators A₁, A₂ with max { ‖ A₁‖ , ‖ A₂‖ } ≤ 2 . We find new characterizations for the Γ -unitaries and describe the distinguished boundary of Γ in a different way. We also show some interplay between the fundamental operators of two Γ -contractions (S, P) and (S₁, P) .