论文标题
toral收缩和$γ$ disting的$γ$ - 结合
The toral contractions and $Γ$-distinguished $Γ$-contractions
论文作者
论文摘要
我们考虑Bidisc $ \ MATHBB d^2 $中的特殊品种和对称的Bidisc $ \ Mathbb {G} _2 $。 $γ= \ overline {\ mathbb {g}} _ 2 $的通勤对$（s，p）$是一个频谱集，称为a \ textit {$γ$ -Contraction}。如果有\ mathbb c [z_1，z_2] $，则一对通勤的收缩$（t_1，t_2）$被称为\ textit {toral}，以便$ p（t_1，t_2，t_2）= 0 $。同样，如果$（s，p）$在$ \ mathbb c [z_1，z_2] $中nikeed灭$（s，p）$，则称为$γ$ -Contraction $（s，p）$，称为$γ$ - \ textit {dectionit {dectine}。本文的主要结果是：我们找到了必要且充分的条件，使得一对收缩会扩张到一对异构体。此外，我们表征了所有$γ$ distance的$γ$ - 承认$γ$ distancessine $γ$γ$ - 等级膨胀的结合。我们确定$ \ Mathbb d^2 $和$ \ Mathbb g_2 $中杰出品种的杰出边界。然后，我们证明一对通勤收缩$（t_1，t_2）$（或$γ$ - 征服$（s，p）$）扩张到一对异构体（或$γ$变化的$γ$ -isometry）时，并且只有在是否存在toral poldnomial（或有$γ$ $γ$ - pol pol pol pol pol pol pol pol pot）， c [z_1，z_2] $使得$ z（p）\ cap \ overline {\ mathbb d}^2 $（或$ z（p）\ cap \ cap \ overline {\ mathbb g} _2 $）是$（t_1，t_1，t_2）$（或$）的完整频谱集。我们在位置提供了几个例子，以显示Toral收缩理论与$γ$ disting的$γ$合同之间的对比。
We consider distinguished varieties in the bidisc $\mathbb D^2$ and the symmetrized bidisc $\mathbb{G}_2$. A commuting pair of Hilbert space operators $(S,P)$ for which $Γ=\overline{\mathbb{G}}_2$ is a spectral set is called a \textit{$Γ$-contraction}. A commuting pair of contractions $(T_1,T_2)$ is said to be \textit{toral} if there is a toral polynomial $p\in \mathbb C[z_1,z_2]$ such that $p(T_1,T_2)=0$. Similarly, a $Γ$-contraction $(S,P)$ is called $Γ$-\textit{distinguished} if $(S,P)$ is annihilated by a $Γ$-distinguished polynomial in $\mathbb C[z_1,z_2]$. The main results af this article are the following: we find a necessary and sufficient condition such that a toral pair of contractions dilates to a toral pair of isometries. Also, we characterize all $Γ$-distinguished $Γ$-contractions that admit $Γ$-distinguished $Γ$-isometric dilations. We determine the distinguished boundary of a distinguished variety in $\mathbb D^2$ and $\mathbb G_2$. Then we prove that a pair of commuting contractions $(T_1,T_2)$ (or a $Γ$-contraction $(S,P)$) dilates to a toral pair of isometries (or a $Γ$-distinguished $Γ$-isometry) if and only if there is a toral polynomial (or $Γ$-distinguished polynomial) $p\in \mathbb C[z_1,z_2]$ such that $Z(p) \cap \overline{\mathbb D}^2$ (or $Z(p) \cap \overline{\mathbb G}_2$) is a complete spectral set for $(T_1,T_2)$ (or $(S,P)$). We provide several examples at places to show the contrasts between the theory of toral contractions and $Γ$-distinguished $Γ$-contractions.
