The symmetrization map and $$\Gamma$$-contractions

Abstract

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