Hermite全纯向量丛的若尔当分解和Kaplansky第二检验问题
The Jordan decomposition and Kaplansky's second test problem for Hermitian holomorphic vector bundles
摘要：
1954年，我. Kaplansky在他的论文“Infinite Abelian groups“中提出了三个测试问题，用于确定对一类数学对象的结构理解的强度，这可以为非常一般的数学系统制定。在本文中，我们专注于Kaplansky的第二个测试问题的背景下，复杂的几何。设$H^2_β$是加权Hardy空间. Cowen-Douglas算子理论告诉我们，每个$h\in\textrm{Hol}(\overline{\mathbb{D}})$在$H^2_β$上诱导一个厄米特全纯向量丛，记为$E_{h(S_β)}(Ω)$，其中$Ω$是一个域。证明了向量丛$E_{h(S_β)}$是一个向前推进的Hermite全纯向量丛，并研究了向量丛的相似形变问题。本文的主要定理是：如果$H^2_β$是多项式增长的加权Hardy空间，则对任何$f\in \textrm{Hol}(\overline{\mathbb{D}})$，存在唯一的正整数$m$和诱导不可分解向量丛$E_{h(S_β)}$的函数$h\in\textrm{Hol}(\overline{\mathbb{D}})$，使得$E_{f(S_β)}$类似于$\bigoplus_1^m E_{h(S_β)}$，其中$h$在解析自同构群作用意义下是唯一的.这可以看作是向前推进埃尔米特全纯向量丛的一个若尔当分解定理。进一步，我们给出了解析函数诱导的向前推进Hermite全纯向量丛的相似分类，并对Kaplansky第二检验问题给出了肯定的回答.对R. Douglas在2007年提出的方法，得到了多项式增长的加权Hardy空间上乘法算子的交换代数的$K_0$ -群.此外，我们还举例说明了多项式增长条件的设置是必要的.
英文摘要：
In 1954, I. Kaplansky proposed three test problems for deciding the strength of structural understanding of a class of mathematical objects in his treatise "Infinite abelian groups", which can be formulated for very general mathematical systems. In this paper, we focus on Kaplansky's second test problem in a context of complex geometry. Let $H^2_β$ be a weighted Hardy space. The Cowen-Douglas operator theory tells us that each $h\in\textrm{Hol}(\overline{\mathbb{D}})$ induces a Hermitian holomorphic vector bundle on $H^2_β$, denoted by $E_{h(S_β)}(Ω)$, where $Ω$ is a domain. We show that the vector bundle $E_{h(S_β)}$ is a push-forwards Hermitian holomorphic vector bundle and study the similarity deformation problems. Our main theorem is that if $H^2_β$ is a weighted Hardy space of polynomial growth, then for any $f\in \textrm{Hol}(\overline{\mathbb{D}})$, there exists a unique positive integer $m$ and an function $h\in\textrm{Hol}(\overline{\mathbb{D}})$ inducing an indecomposable vector bundle $E_{h(S_β)}$, such that $E_{f(S_β)}$ is similar to $\bigoplus_1^m E_{h(S_β)}$, where $h$ is unique in the sense of analytic automorphism group action. That could be seemed as a Jordan decomposition theorem for the push-forwards Hermitian holomorphic vector bundles. Furthermore, we give the similarity classification of those push-forwards Hermitian holomorphic vector bundles induced by analytic functions, and give an affirmative answer to Kaplansky's second test problem for those objects. We also give an affirmative answer to the geometric version and generalized version of a problem proposed by R. Douglas in 2007, and obtain the $K_0$-group of the commutant algebra of a multiplication operator on a weighted Hardy space of polynomial growth. In addition, we give an example to show the setting of polynomial growth condition is necessary.
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