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 $(S_1,P)$.