论文标题
在对称的bidisc上的Carathéodory极端功能的典范性
Nonuniqueness of Carathéodory extremal functions on the symmetrized bidisc
论文作者
论文摘要
我们调查了对对称的bidisc $$ g = \ = \ = \ {（z+w，zw）：| z | <1，\，| w | <1 \} = \ {（s，s，s，s，s，s，s，s c} c} c} c}^2：s p | <1- | p |^2 \}。 $$我们还为此主题提供了一些新的结果。我们对问题解决方案不是唯一的问题特别感兴趣。众所周知，对于任何$δ=（λ，v）\ in tg $ in tg $ with $ v \ neq 0 $，至少有一个$ω\ in \ mathbb {t} $中的一个$ω\，以便$φ_Ω$ solves $ \ mathrm {car}Δ$ solves $ \ mathrm {car}δ$，$ q $ q（s，p）$φ_Ω此外，当$ \ mathrm {car}δ$的本质上是独特的解决方案，并且仅当$δ$完全具有一个carathéodory的极端函数$φ_Ω$的一个carathéodory极函数。我们对Tg $中的$δ\的Carathéodory极端分子进行描述，其中一个以上的carathéodory极函数$φ_Ω$，对于某些值$ω\ in \ mathbb {t} $。该证明利用了$ g $的Schur类别的模型公式，这是对光盘上Schur-Class函数的著名网络实现公式的类似物。
We survey the Carathéodory extremal problem $\mathrm{Car} δ$ on the symmetrized bidisc $$ G = \{(z+w,zw):|z|<1, \, |w|<1\} = \{(s,p)\in \mathbb{C}^2: |s-\bar s p| < 1-|p|^2\}. $$ We also give some new results on this topic. We are particularly interested in cases of this problem in which the solution of the problem is not unique. It is known that, for any $δ=(λ,v)\in TG$ with $v\neq 0$, there is at least one $ω\in\mathbb{T}$ such that $Φ_ω$ solves $\mathrm{Car} δ$, where $Φ_ω(s,p) = \frac{2ωp-s}{2-ωs}$. Moreover, there is an essentially unique solution of $\mathrm{Car} δ$ if and only if $δ$ has exactly one Carathéodory extremal function of the form $Φ_ω$ for some $ω\in\mathbb{T}$. We give a description of Carathéodory extremals for $δ\in TG$ with more than one Carathéodory extremal function $Φ_ω$ for some values of $ω\in\mathbb{T}$. The proof exploits a model formula for the Schur class of $G$ which is an analog of the well-known network realization formula for Schur-class functions on the disc.
