Operators on reproducing kernel Banach spaces of analytic functions

提议的研究项目是在数学分析领域。它的总体目标是提高算子理论和复分析的知识，以及它们之间的相互作用。这些领域与一些自然科学和工程学有着密切的联系，在大多数情况下，这些问题是由特定的应用直接引起的。在经典分析中，函数是研究的主要对象，而现代分析的目标是研究函数类的变换。本文拟研究的一类函数是解析函数的再现核巴拿赫和希尔伯特空间，而变换是由这些空间的结构自然决定的线性算子。这些空间的一些著名的和被广泛探索的例子是Bergman， Hardy, Dirichlet， Bloch和Besov空间。所讨论的运算符类包括乘法、复合、Toeplitz、积分和条件期望运算符。这类空间的一个很好的性质是我们可以使用再现核来计算空间中在其定义域的特定点上的函数。当我们试图从采样和测量的实验过程中尽可能准确地重现一个函数时，我们自然会这样做。在这种情况下，一个特别有趣的问题是，通过知道一个算子在再现核函数空间上的行为，进一步确定我们能说多少关于作用于再现核函数空间的算子的性质。因此，所提出的程序的更具体的目标是通过反映它们所作用的再现核函数空间的结构来分类和确定自然定义的一大类算子的性质，同时也对空间本身有更深入的了解。***提出的研究计划的科学方法使用了现代和经典数学分析的几个领域的方法。除了标准算子理论和复分析技术外，它还涉及到一般测度理论、几何函数理论和线性代数方法。***我的团队方法的新颖之处在于，我们只提取了空间和待探索算子的几个基本必需属性，从而尝试推广处理特定空间上特定算子的几个最新分类结果。与我的HQP一起，我希望推导出一个模型，一方面解决更一般性质的问题，另一方面提供对探索对象结构的更深入的洞察。除了数学之外，这些类型的结果在自然科学和工程的其他领域也有直接的应用，如量子力学、量子信息、控制论、机器学习、图像处理和统计学。

The proposed research program is in the area of mathematical analysis. Its general objective is the enhancement of knowledge in operator theory and complex analysis, and of the interactions between them. These are areas with close connections to several natural sciences and to engineering, and in most of the cases the problems are directly motivated by specific applications.***While functions are the main objects of exploration in classical analysis, the goal of modern analysis is to explore the transformations of classes of functions. The class of functions we plan to investigate in this proposal is the reproducing kernel Banach and Hilbert spaces of analytic functions, while the transformations are the linear operators determined naturally by the structure of these spaces. Some of the well-known and widely explored examples of such spaces are the Bergman, Hardy, Dirichlet, Bloch and Besov spaces. The classes of operators in question include the multiplication, composition, Toeplitz, integral and conditional expectation operators.***A nice property of these types of spaces is that we can use the reproducing kernels to evaluate the functions in the space at a specific point of their domain. This is what we naturally do when we attempt to reproduce a function, as accurately as possible, from an experimental process of sampling and measuring. A particularly interesting question in this context is to furthermore determine how much we can say about the properties of an operator acting on a reproducing kernel function space, by knowing how it behaves on the reproducing kernel functions. Hence, the more specific objective of the proposed program is to classify and determine the properties of a large class of operators defined naturally by reflecting the structure of the reproducing kernel function spaces that they act on, while at the same time also gaining a deeper understanding of the spaces themselves.***The scientific approach of the proposed research program uses methods from several areas of modern and classical mathematical analysis. Beside the standard operator theoretic and complex analysis techniques, it also involves general measure theory, geometric function theory and linear algebra methods. ***The novelty in my teams approach is that we extract only few basic required properties of the spaces and the operators to be explored, and thus attempt to generalize several of the more recent classification results dealing with specific operators on specific spaces. Together with my HQP', I hope to derive a model which on one hand addresses problems of more general nature, and on the other, provides a deeper insight into the structure of the objects of exploration. Beside mathematics, these types of results are of interest and have direct applications in other areas of natural sciences and engineering such as quantum mechanics, quantum information, control theory, machine learning, image processing and statistics.