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,

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 \((S_1,P)\).

对称映射\(\pi :{\mathbb{C}}^2\rightarrow {\mathbb{C}}^2\)由\(\pi (z_1,z_2)=(z_1+z_2,z_1z_2) 定义.\)闭对称 bidisc \(\Gamma\)是闭单元 bidisc \(\overline{{\mathbb{D}}^2}\)的对称化，即

一对交换希尔伯特空间算子 ( S ,  P )，其中\(\Gamma\)是一个谱集，称为\(\Gamma\) -收缩。与 \(\Gamma\) 中的标量不同，\(\Gamma\) -收缩可能不会作为一对通勤收缩的对称化出现，甚至不是作为一对通勤有界算子的对称化。我们描述了所有的\(\Gamma\) - 收缩，它们是通勤收缩对的对称化。我们通过构建一系列示例表明，即使对于一对通勤有界算子\ ( T_1, T_2 \ )，一般来说，没有小于 2 的实数可以成为集合\(\{ \Vert T_1\Vert ,\Vert T_2\Vert \}\)的界。然后我们证明每个\(\Gamma\) -contraction ( S ,  P ) 是一个\(\Gamma\) -contraction \(({{\widetilde{S}}}, {{\widetilde{P }}})\)到\({{\widetilde{S}}}, {{\widetilde{P}}}\)和\(({{\widetilde{S}}}, {{\widetilde{P}}})=(A_1+A_2,A_1A_2)\)用于一对通勤运算符\(A_1,A_2\)与\(\max \{\Vert A_1\Vert , \Vert A_2\垂直 \} \le 2\)。我们发现了\(\Gamma\)的新特征-unitaries 并以不同的方式描述\(\Gamma\)的可区分边界。我们还展示了两个\(\Gamma\) -收缩（S ,  P）和\((S_1,P)\)的基本算子之间的一些相互作用。