论文标题
Schwarz引理五角杆的引理
A Schwarz lemma for the pentablock
论文作者
论文摘要
在本文中，我们证明了五边形的施瓦茨引理。集合\ [\ Mathcal {p} = \ {（a_ {21}，\ text {tr} \ a，\ det a）：a = [a_ {a_ {ij}] _ {i，j = 1}^2 \ in \ mathbb {b} $ \ mathbb {b}^{2 \ times 2} $表示在$ 2 \ times 2 $复杂矩阵的空间中的打开单位球，称为pentablock。 Pentablock是$ \ bbb {c}^3 $中有界的非convex域，它与$μ$ - 同时的某些问题自然产生。我们开发了一种从单位光盘$ \ bbb {d} $的理性映射到关闭的pentablock $ \ overline {\ Mathcal {p}} $的具体结构理论$ \ OVILLINE {\ MATHCAL {P}} $。这样的地图称为Rational $ {\ Overline {\ Mathcal {p}}}} $ - 内函数。我们将PENTA INNER函数与内部功能从$ \ bbb {d} $到对称的Bidisc之间的关系。我们描述了有理penta-Inner函数的构建$ x =（a，s，p）：\ bbb {d} \ rightArrow \ edrow \ overline {\ mathcal {p}} $从$ a，s $和$ s $ and $ s $ and $ s^2-4p $的零。该定理的证明是建设性的：它给出了一种算法，用于构建此类功能的家族$ x $，但要根据圆圈上某些非负三角函数的Fejér-riesz fracesivation计算。我们使用属性和构建Rational $ {\ overline {\ Mathcal {p}}} $ - 内部函数来证明Pentablock的Schwarz引理。
In this paper we prove a Schwarz lemma for the pentablock. The set \[ \mathcal{P}=\{(a_{21}, \text{tr} \ A, \det A) : A=[a_{ij}]_{i,j=1}^2 \in \mathbb{B}^{2\times 2}\} \] where $\mathbb{B}^{2\times 2}$ denotes the open unit ball in the space of $2\times 2$ complex matrices, is called the pentablock. The pentablock is a bounded nonconvex domain in $\Bbb{C}^3$ which arises naturally in connection with a certain problem of $μ$-synthesis. We develop a concrete structure theory for the rational maps from the unit disc $\Bbb{D}$ to the closed pentablock $\overline{\mathcal{P}}$ that map the unit circle ${\mathbb{T}}$ to the distinguished boundary $b\overline{\mathcal{P}}$ of $\overline{\mathcal{P}}$. Such maps are called rational ${\overline{\mathcal{P}}}$-inner functions. We give relations between penta-inner functions and inner functions from $\Bbb{D}$ to the symmetrized bidisc. We describe the construction of rational penta-inner functions $x = (a, s, p) : \Bbb{D} \rightarrow \overline{\mathcal{P}}$ of prescribed degree from the zeroes of $a, s$ and $s^2-4p$. The proof of this theorem is constructive: it gives an algorithm for the construction of a family of such functions $x$ subject to the computation of Fejér-Riesz factorizations of certain non-negative trigonometric functions on the circle. We use properties and the construction of rational ${\overline{\mathcal{P}}}$-inner functions to prove a Schwarz lemma for the pentablock.
