Resumen de The symmetrization map and T-contractions

Sourav Pal

• The symmetrization map $$\pi :{\mathbb{C}}^2\rightarrow {\mathbb{C}}^2$$ π : C 2 → C 2 is defined by $$\pi (z_1,z_2)=(z_1+z_2,z_1z_2).$$ π ( z 1 , z 2 ) = ( z 1 + z 2 , z 1 z 2 ) . The closed symmetrized bidisc $$\Gamma$$ Γ is the symmetrization of the closed unit bidisc $$\overline{{\mathbb{D}}^2}$$ D 2 ¯ , that is, $$\begin{aligned} \Gamma = \pi (\overline{{\mathbb{D}}^2})=\{ (z_1+z_2,z_1z_2)\,:\, |z_i|\le 1, i=1,2 \}. \end{aligned}$$ 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 $$\Gamma$$ Γ is a spectral set is called a $$\Gamma$$ Γ -contraction. Unlike the scalars in $$\Gamma$$ Γ , a $$\Gamma$$ 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 $$\Gamma$$ Γ -contractions which are symmetrization of pairs of commuting contractions. We show by constructing a family of examples that even if a $$\Gamma$$ Γ -contraction $$(S,P)=(T_1+T_2,T_1T_2)$$ ( S , P ) = ( T 1 + T 2 , T 1 T 2 ) for a pair of commuting bounded operators $$T_1,T_2$$ T 1 , T 2 , no real number less than 2 can be a bound for the set $$\{ \Vert T_1\Vert ,\Vert T_2\Vert \}$$ { ‖ T 1 ‖ , ‖ T 2 ‖ } in general. Then we prove that every $$\Gamma$$ Γ -contraction (S, P) is the restriction of a $$\Gamma$$ Γ -contraction $$({{\widetilde{S}}}, {{\widetilde{P}}})$$ ( S ~ , P ~ ) to a common reducing subspace of $${{\widetilde{S}}}, {{\widetilde{P}}}$$ S ~ , P ~ and that $$({{\widetilde{S}}}, {{\widetilde{P}}})=(A_1+A_2,A_1A_2)$$ ( S ~ , P ~ ) = ( A 1 + A 2 , A 1 A 2 ) for a pair of commuting operators $$A_1,A_2$$ A 1 , A 2 with $$\max \{\Vert A_1\Vert , \Vert A_2\Vert \} \le 2$$ max { ‖ A 1 ‖ , ‖ A 2 ‖ } ≤ 2 . We find new characterizations for the $$\Gamma$$ Γ -unitaries and describe the distinguished boundary of $$\Gamma$$ Γ in a different way. We also show some interplay between the fundamental operators of two $$\Gamma$$ Γ -contractions (S, P) and $$(S_1,P)$$ ( S 1 , P ) .