Blaschke乘积、水平集和Crouzeix猜想
Blaschke Products, Level Sets, and Crouzeix's Conjecture
摘要：
我们研究了Crouzeix猜想的动机，我们认为在特殊的设置的模型空间和压缩的移位有限Blaschke产品作为符号的几个问题。我们提出了一个版本的猜想在这种情况下，称为水平集Crouzeix（LSC）猜想，并建立结构和唯一性（开放）水平集的有限Blaschke产品，使我们能够证明LSC猜想在几种情况下。特别是，我们使用的数值域的几何证明的LSC猜想压缩的移位对应的单临界Blaschke产品的程度$4$。
英文摘要：
We study several problems motivated by Crouzeix's conjecture, which we consider in the special setting of model spaces and compressions of the shift with finite Blaschke products as symbols. We pose a version of the conjecture in this setting, called the level set Crouzeix (LSC) conjecture, and establish structural and uniqueness properties for (open) level sets of finite Blaschke products that allow us to prove the LSC conjecture in several cases. In particular, we use the geometry of the numerical range to prove the LSC conjecture for compressions of the shift corresponding to unicritical Blaschke products of degree $4$.
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