The symmetrization map and Γ-contractions.

The symmetrization map π : C 2 → C 2 is defined by π (z 1 , z 2) = (z 1 + z 2 , z 1 z 2). The closed symmetrized bidisc Γ is the symmetrization of the closed unit bidisc D 2 ¯ , that is, Γ = π ( D 2 ¯) = { (z 1 + z 2 , z 1 z 2) : | z i | ≤ 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) = (T 1 + T 2 , T 1 T 2) for a pair of commuting bounded operators T 1 , T 2 , no real number less than 2 can be a bound for the set { ‖ T 1 ‖ , ‖ T 2 ‖ } 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 ~) = (A 1 + A 2 , A 1 A 2) for a pair of commuting operators A 1 , A 2 with max { ‖ A 1 ‖ , ‖ A 2 ‖ } ≤ 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 (S 1 , P) . [ABSTRACT FROM AUTHOR]