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

The symmetrization map $$\pi :{\mathbb{C}}^2\rightarrow {\mathbb{C}}^2$$ is defined by (z_1,z_2)=(z_1+z_2,z_1z_2).$$ closed symmetrized bidisc $$\Gamma$$ the of unit $$\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}$$ A pair commuting Hilbert space operators (S, P) for which a spectral set called -contraction. Unlike scalars in -contraction may not arise as contractions, even bounded operators. We characterize all -contractions are pairs contractions. show constructing family examples if $$(S,P)=(T_1+T_2,T_1T_2)$$ $$T_1,T_2$$ no real number less than 2 can be bound $$\{ \Vert T_1\Vert ,\Vert T_2\Vert \}$$ general. Then we prove every restriction $$({{\widetilde{S}}}, {{\widetilde{P}}})$$ to common reducing subspace $${{\widetilde{S}}}, {{\widetilde{P}}}$$ and {{\widetilde{P}}})=(A_1+A_2,A_1A_2)$$ $$A_1,A_2$$ with $$\max \{\Vert A_1\Vert A_2\Vert \} \le 2$$ . find new characterizations -unitaries describe distinguished boundary different way. also some interplay between fundamental two $$(S_1,P)$$