论文标题
与光谱插值和操作者理论相关的域家族中的杰出品种
Distinguished varieties in a family of domains associated with spectral interpolation and operator theory
论文作者
论文摘要
我们发现在对称的polydisc $ \ mathbb g_n \中的杰出品种的表征; （n \ geq 2）$，从而概括了工作[\ textit {j。功能。肛门。}，266（2014），5779-5800]在作者和Shalit上$ \ Mathbb g_2 $。我们表明，$ \ mathbb g_n $中的杰出品种$λ$是代数曲线的一部分，该曲线是一个固定的理论完整交叉点，$λ$可以由$ n-1 $的泰勒关节频谱代表$ n-1 $通勤标量的标量矩阵满足某些条件。 $ n $ - 通勤Hilbert太空运营商$（s_1，\ dots，s_ {n-1}，p）$，$γ_n= \ overline {\ mathbb g_n} $是一个频谱集，称为$γ_n$ -contraction。对于每个$γ_n$ -Contraction $（s_1，\ dots，s_ {n-1}，p）$，都有一个唯一的操作员元组$（f_1，\ dots，f_ {n-1}）$，称为$ \ Mathcal f_o $ $ $ $ $ $ - $（ s_i-s_ {n-i}^*p = d_pf_id_p \ ,, \ quad i = 1，\ dots，n-1。 \]我们为与$γ_n$相关的纯等轴测操纵子元组生成具体的功能模型，并且通过该模型的应用，我们确定$γ_n$ - 汇总$（s_1，\ dots，s_ {n-1}，s_ {n-1}，p），p）$ and $（s_1^*，s_1^*，s_1^*，\ dots，s_ n-1-1} \ overline {λ}_σ-$扩展是独特的区分品种$ \ mathbb g_n $中的$λ_σ$，当$ \λ_σ$由$ \ mathcal f_o $ -tuple $（s_1，\ dots，s_ {n-1}，p）$确定。此外，我们表明$（s_1^*，\ dots，s_ {n-1}^*，p^*）$的扩张是最小的，并且在$ p^*$的最小统一扩张空间上作用。另外，我们在$ \ mathbb g_2 $和$ \ mathbb g_ {3} $中显示了杰出品种之间的相互作用。
We find characterization for the distinguished varieties in the symmetrized polydisc $\mathbb G_n \; (n\geq 2)$ and thus generalize the work [\textit{J. Funct. Anal.}, 266 (2014), 5779 -- 5800] on $\mathbb G_2$ by the author and Shalit. We show that a distinguished variety $Λ$ in $\mathbb G_n$ is a part of an algebraic curve, which is a set-theoretic complete intersection, and that $Λ$ can be represented by the Taylor joint spectrum of $n-1$ commuting scalar matrices satisfying certain conditions. An $n$-tuple of commuting Hilbert space operators $(S_1, \dots ,S_{n-1},P)$ for which $Γ_n=\overline{\mathbb G_n}$ is a spectral set is called a $Γ_n$-contraction. To every $Γ_n$-contraction $(S_1, \dots ,S_{n-1},P)$ there is a unique operator tuple $(F_1, \dots , F_{n-1})$, called the $\mathcal F_O$-tuple of $(S_1, \dots ,S_{n-1},P)$, satisfying \[ S_i-S_{n-i}^*P=D_PF_iD_P \,,\quad i=1, \dots ,n-1. \] We produce concrete functional model for the pure isometric-operator tuples associated with $Γ_n$ and by an application of that model we establish that the $Γ_n$-contractions $(S_1, \dots ,S_{n-1},P)$ and $(S_1^*, \dots , S_{n-1}^*,P^*)$ admit normal $\partial \overline{ Λ}_Σ-$dilations for a unique distinguished variety $Λ_Σ$ in $\mathbb G_n$, when $Λ_Σ$ is determined by the $\mathcal F_O$-tuple of $(S_1, \dots ,S_{n-1}, P)$. Further, we show that the dilation of $(S_1^*, \dots ,S_{n-1}^*,P^*)$ is minimal and acts on the minimal unitary dilation space of $P^*$. Also, we show interplay between the distinguished varieties in $\mathbb G_2$ and $\mathbb G_{3}$.
