A Schwarz lemma for the symmetrized tridisc and description of interpolating functions

对称三圆盘上的有理扩张：失败，成功和未知

Rational dilation on the symmetrized tridisc: failure, success and unknown

闭对称化三圆盘$Γ_3$和它的区别边界$bΓ_3$是集合$Γ_3=\{ (z_1+z_2+z_3,z_1z_2+z_2z_3+z_3z_1,z_1z_2z_3): \,|z_i|\leq 1, i=1,2,3 \}\subseteq \mathbb C^3$ $bΓ_3=\{ (z_1+z_2+z_3,z_1z_2+z_2z_3+z_3z_1,z_1z_2z_3): \,|z_i|= 1, i=1,2,3 \}\subseteq Γ_3.$定义在希尔伯特空间$\mathcal H$上的交换算子$(S_1,S_2,P)$的三元组，其中$Γ_3$是其谱集，称为$Γ_3$ -压缩。在这篇文章中，我们通过一个反例表明，存在不扩张的$Γ_3$ -收缩。还表明在某些条件下，$Γ_3$收缩可以具有正常的$bΓ_3$扩张。我们确定了几个类的$Γ_3$ -收缩扩张，并显示其扩张明确的建设。为扩张的$Γ_3$ -收缩提供了一个具体的功能模型。得到了$Γ_3$ -酉和$Γ_3$ -等距的各种刻画; $Γ_3$ -酉和$Γ_3$ -等距的类类似于一元算子理论中的酉和等距。我们还找到了纯$Γ_3$ -等距类的一个模型。在途中，我们研究了集合$Γ_3$和$bΓ_3$的几何，并为它们提供了各种特征。

算子理论与对称三圆盘上的区别簇表示

Operator theory and representation of distinguished varieties in the symmetrized tridisc

我们证明了对称化三圆盘$\mathbb G_3$中的每一个特殊簇都是一维的，并且可以表示为\begin{equation}\label{eqn:1} Λ=\{ (s_1,s_2,p)\in \mathbb G_3 \,:\, (s_1,s_2) \in σ_T(F_1^*+pF_2\,,\, F_2^*+pF_1) \}, \end{equation}，其中$F_1,F_2$是满足$[F_1^*,F_1]=[F_2^*,F_2]$和一个范数条件的同阶交换方阵.反之亦然，即一个形式为（\ref{eqn:1}）的集合总是$\mathbb G_3$中的一个特殊变种。本文证明了对于以$Γ_3$为谱集的交换算子三元组$Σ= (S_1,S_2,P)$，存在$Γ_3$的一维子簇$Λ_Σ$依赖于$Σ$，使得von-Neumann不等式成立，即对于任何三元全纯多项式$f$，只要$P^n\rightarrow 0$强如$n\rightarrow \infty$，就有\[ f(S_1,S_2,P)\leq \sup_{(s_1,s_2,p)\inΛ_Σ}\, |f(s_1,s_2,p)|, \].证明了簇$Λ_Σ$具有类似（\ref{eqn:1}）的表示，其中$F_1,F_2$是算子方程\begin{gather*} S_1-S_2^*P=(I-P^*P)^{\frac{1}{2}}X_1(I-P^*P…

对称双圆盘边界上的Schwarz引理

The Schwarz Lemma at the Boundary of the Symmetrized Bidisc

对称双圆盘${\textbf{G}}_{2}$由$${\textbf{G}}_{2}:=\{(z_1+z_2,z_1z_2)\in\mathbb{C}^2: |z_1|<1,|z_2|<1,\; z_1,z_2\in\mathbb{C}\}.$$定义。它是一个没有$\mathcal{C}^1$边界的有界非齐次拟凸域，特别是对称双圆盘上没有强拟凸边界点，Carathéodory度量和Kobayashi度量在对称双圆盘上的边界表现很难精确描述。本文研究了对称双圆盘${\textbf{G}}_2$上全纯自映射的边界Schwarz引理，所得的边界Schwarz引理与以前的相关结果有很大的不同.

A Schwarz lemma for two families of domains and complex geometry

We make sharp estimates to obtain a Schwarz type lemma for the symmetrized polydisc $\gn$ and for the extended symmetrized polydisc $\Gn$. We explicitly construct an interpolating function under certain condition. To do so, we followed the methods described in \cite{Young-LMS}. Also we find a few geometric interplay between the members of the family $\Gn$ and its closure $\widetildeΓ_n$.

A Schwarz lemma for the symmetrized polydisc via estimates on another family of domains

We make some sharp estimates to obtain a Schwarz lemma for the \textit{symmetrized polydisc} $\mathbb G_n$, a family of domains naturally associated with the spectral interpolation, defined by \[ \mathbb G_n :=\left\{ \left(\sum_{1\leq i\leq n} z_i,\sum_{1\leq i<j\leq n}z_iz_j \dots, \prod_{i=1}^n z_i \right): \,|z_i|<1, i=1,\dots,n \right \}. \] We first make a few estimates for the \textit{the extended symmetrized polydisc} $\widetilde{\mathbb G}_n$, a family of domains introduced in \cite{pal-roy 4} and defined in the following way: \begin{align*} \widetilde{\mathbb G}_n := \Bigg\{ (y_1,\do…

A generalized Schwarz lemma for two domains related to $μ$-synthesis

We present a set of necessary and sufficient conditions that provides a Schwarz lemma for the tetrablock $\mathbb E$. As an application of this result, we obtain a Schwarz lemma for the symmetrized bidisc $\mathbb G_2$. In either case, our results generalize all previous results in this direction for $\mathbb E$ and $\mathbb G_2$.