臺灣博碩士論文加值系統

在此篇論文中吾人討論在對稱雙盤$\Gamma$上之Schwarz
引理及$\Gamma$上之特殊平的問題如下:
給定$\alpha_{2}\in\mathbb{D},~\alpha_{2}\neq0~$
且 $(s_{2},p_{2})\in\Gamma$, 找一個解析函數
$\varphi:\mathbb{D}\rightarrow\Gamma~~$使得
$\varphi(\lambda)=(s(\lambda),p(\lambda))~~~$ 且滿足
$$\varphi(0)=(0,0),~\varphi(\alpha_{2})=(s_{2},p_{2})$$
由Carath\'{e}odory與Kobayashi距離相等的條件下,
及Schur's定理吾人成功的建構出一個滿足此問題的解析函數$\varphi$.
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關鍵字:Spectral Nevanlinna-Pick 插值, Poincar\'{e} 距離,
Carath\'{e}odory 距離, Kobayashi 距離,對稱雙盤, Schwarz 引理.

Let $\Gamma$ denote the set of symmetrized bidisc. In this thesis
we discuss the Schwarz lemma on $\Gamma$ also known as the special
flat problem on $\Gamma$ as:
Given
$\alpha_{2}\in\mathbb{D},~\alpha_{2}\neq0~$
and $(s_{2},p_{2})\in\Gamma$, find an analytic function
$\varphi:\mathbb{D}\rightarrow\Gamma$with
$\varphi(\lambda)=(s(\lambda),p(\lambda))$ satisfies
$$\varphi(0)=(0,0),~\varphi(\alpha_{2})=(s_{2},p_{2})$$
Based on the equality of Carath\'{e}odory and Kobayashi distances,
and the Schur's theorem, we construct an analytic function
$\varphi$ to solve this problem.
Keywords: Spectral Nevanlinna-Pick interpolation,
Poincar\'{e} distance, Carath\'{e}odory distance, Kobayashi
distance, Symmetrized bidisc, Schwarz lemma.

1 Introduction 1
1.1 Notation 1
1.2 Motivation 3
2.Mathematical Preliminaries 11
3.Main Results 14
3.1 Some properties of $\Gamma$ 14
3.2 The ideal to construct $\varphi$ 20
3.3 Realization of Symmetrized Bidisc $\Gamma$ 25
4 Concluding Remarks 33
Appendex 34
References 36

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