Mathematical Sciences: Subnormal Operators and Banach Spaces of Holomorphic Functions

项目摘要（C项）项目负责人将继续研究次正规算子的内部结构理论，特别考虑有限秩自对易子、不变子空间的结构以及在其他领域的联系和应用。他将使用算子理论工具来研究正交域的理论。他还计划研究单位盘上Bergman空间的不变子空间的结构以及一般有界区域上Bergman空间之间的乘子。算子理论是现代数学中一门研究矩阵无限维泛化的核心学科。这种类型的研究是试图对满足附加条件的算子进行分类，这些附加条件是根据相关算子或基础空间给出的。它们不仅在纯数学中，而且在每一门应用科学中都有丰富的应用。举一个具体的例子，一个基本的h -∞控制问题是由Nevanlinna-Pick插值的经典数学理论提出的。该理论的应用包括改进汽车悬架系统、飞机飞行控制、化工过程控制等。

Abstract of the project (item C) The principal investigator will continue to study the internal structure theory of subnormal operators, with particular consideration of finite rank self-commutator, the structure of invariant subspaces, and connections and applications to other fields. He will use operator theoretical tools to study the theory of quadrature domains. He also plans to investigate the structure of invariant subspaces of the Bergman space on the unit disk and multipliers between the Bergman spaces over a general bounded region. Operator theory is a central discipline in modern mathematics that studies infinite dimensional generalization of matrices. This type of research is an attempt to classify operators satisfying additional conditions given in terms of the associated operator or the underlying space. They have rich applications in every applied science as well as pure mathematics. To mention a specific example, one of basic H-infinity control problems was developed by the classical mathematical theory of Nevanlinna-Pick interpolation. Applications of the theory include improvement in automobile suspension systems, airplane flight control, chemical process control, etc.