截断Toeplitz算子与Bergman空间上的乘法算子

发布时间：2018-05-21 01:20

本文选题：模型空间 + 截断Toeplitz算子　；参考：《重庆大学》2016年博士论文

【摘要】：算子理论是泛函分析的重要组成部分.作为是算子理论的重要分支,解析函数空间上的算子理论,一直得到国内外学者的持续关注.因为Toeplitz算子理论是解析函数空间上的一类非常重要的算子,所以Toeplitz算子的研究虽然已经超过半个世纪,取得了大量的成果,但直到今天Toeplitz算子的相关研究依然很活跃.主要的原因包括下面两个方面:一方面,Toeplitz算子与von Neumann代数、非交换几何、随机矩阵、量子信息、工程控制理论等有密切的关系;另外一方面,研究函数空间上的Toeplitz算子和Toeplitz代数无论是对数学科学本身,还是对物理学以及工程技术的发展都会起着紧要的作用.本文主要研究模型空间(Model space)上的截断Toeplitz算子和Bergman空间上的乘法算子.首先,我们研究了模型空间上的截断Toeplitz算子的紧性.Hardy空间上有界的Toeplitz算子的符号是唯一的,而模型空间上有界的截断Toeplitz算子对应的符号是不唯一的.Baranov,Chalendar,Fricain已经构造出了没有有界符号的有界的截断Toeplitz算子.所以,本文只考虑具有有界符号的截断Toeplitz算子的紧性.我们主要利用Hardy空间上的Hankel算子的乘积和函数代数中极大理想空间的相关技巧,得到了具有有界符号的截断Toeplitz算子的紧性的充分必要条件.这样,Sarason和Bessonov关于截断Toeplitz算子紧性的结果只是这个充分必要条件的特殊情况.其次,我们研究了Bergman空间上的乘法算子生成的von Neumann代数的性质和可交换性.在高维区域上的Bergman空间,考虑以全纯真映射为符号的乘法算子生成的von Neumann代数的性质和可交换性.在一些有趣的情形下,这些性质依赖于一个特殊的黎曼流形.算子理论,几何和复分析在研究中交互出现.最后,我们总结了本学位论文研究的主要结果,并提出本文尚未克服的困难和希望进一步考虑的问题。
[Abstract]:Operator theory is an important part of functional analysis. As an important branch of operator theory, the operator theory in analytic function space has been continuously concerned by scholars at home and abroad. Because Toeplitz operator theory is a very important class of operators on analytic function space, the study of Toeplitz operator has been studied for more than half a century, and a lot of achievements have been made, but the research on Toeplitz operator is still very active up to now. The main reasons are as follows: on the one hand, Toeplitz operators are closely related to von Neumann algebra, noncommutative geometry, random matrix, quantum information, engineering control theory, etc. On the other hand, The study of Toeplitz operators and Toeplitz algebras on function spaces will play an important role in the development of mathematical science, physics and engineering technology. In this paper, the truncated Toeplitz operator on the model space and the multiplication operator on the Bergman space are studied. First, we study the compactness of truncated Toeplitz operators on model spaces. The sign of bounded Toeplitz operators on Hardy spaces is unique. The bounded truncation Toeplitz operator corresponding to the sign is not unique. Baranovi Chalendarn Fricain has constructed a bounded truncated Toeplitz operator without bounded sign. Therefore, we only consider the compactness of truncated Toeplitz operators with bounded symbols. By using the product of Hankel operators on Hardy spaces and the relevant techniques of maximal ideal spaces in function algebra, we obtain a sufficient and necessary condition for the compactness of truncated Toeplitz operators with bounded symbols. So Sarason's and Bessonov's results on truncating the compactness of Toeplitz operators are only a special case of this necessary and sufficient condition. Secondly, we study the properties and commutativity of von Neumann algebras generated by multiplication operators on Bergman spaces. The properties and commutativity of von Neumann algebras generated by multiplicative operators with all pure mappings as symbols are considered in Bergman spaces of high dimensional domains. In some interesting cases, these properties depend on a special Riemannian manifold. Operator theory, geometry and complex analysis interact in the study. Finally, we summarize the main results of this dissertation, and put forward the difficulties which have not been overcome in this paper and the problems we hope to consider further.
【学位授予单位】：重庆大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O177

【参考文献】