Operators on reproducing kernel Banach spaces of analytic functions

解析函数的核Banach空间再现的算子

• 批准号: RGPIN-2017-04975
• 负责人: Zorboska, Nina
• 金额: $ 1.17万
• 依托单位: University of Manitoba
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=675758
• 关键词: Operators; reproducing; kernel; Banach; spaces

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.

拟议的研究计划在数学分析领域。它的总体目标是增强操作者理论和复杂分析的知识以及它们之间的相互作用。这些领域是与几个自然科学和工程的紧密联系的领域，在大多数情况下，这些问题是直接由特定应用的动机。我们计划在此提案中调查的功能类别是分析功能的内核Banach和Hilbert空间，而转换是由这些空间结构自然决定的线性操作员。一些著名且广泛探索的此类空间的例子是伯格曼，哈迪，迪里奇，布洛克和贝斯托夫空间。所讨论的运算符类别包括乘法，组成，Toeplitz，积分和条件期望运算符。***这些类型的空间的一个很好的属性是，我们可以使用复制的内核来评估其域特定点的空间中的功能。当我们尝试从采样和测量的实验过程中重现功能，这是我们自然而然地做的。在这种情况下，一个特别有趣的问题是，通过知道它在复制的内核函数上的作用，可以确定我们对操作员作用在复制内核函数空间上的属性的评价。因此，提出的计划的更具体的目标是通过反映其作用的再现核心函数空间的结构来对大量运营商的特性进行分类和确定，同时也对空间本身进行了更深入的了解。除了标准运算符理论和复杂分析技术外，它还涉及一般测量理论，几何函数理论和线性代数方法。 ***我的团队方法中的新颖性是，我们仅提取空间和要探索的运算符的基本必需属性，因此试图概括几个与特定空间上特定运营商有关的最新分类结果。与我的HQP'一起，我希望得出一个模型，一方面，该模型解决了更一般性的问题，另一方面，它可以更深入地了解探索对象的结构。除了数学外，这些类型的结果令人感兴趣，并在自然科学和工程的其他领域中直接应用，例如量子力学，量子信息，控制理论，机器学习，图像处理和统计数据。