Operators on reproducing kernel Banach spaces of analytic functions

解析函数的核Banach空间再现算子

• 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: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=711326
• 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.

拟议的研究计划属于数学分析领域。其总体目标是增强算子理论和复分析以及它们之间相互作用的知识。这些领域与一些自然科学和工程学密切相关，在大多数情况下，问题是由特定应用直接引发的。 函数是经典分析的主要探索对象，而现代分析的目标是探索函数类的变换。我们计划在本提案中研究的函数类别是解析函数的再现核巴纳赫和希尔伯特空间，而变换是由这些空间的结构自然确定的线性算子。此类空间的一些众所周知且被广泛探索的例子包括伯格曼空间、哈代空间、狄利克雷空间、布洛赫空间和贝索夫空间。所讨论的运算符类别包括乘法、复合、Toeplitz、积分和条件期望运算符。 这些类型的空间的一个很好的特性是，我们可以使用再现内核来评估空间中特定点的函数。当我们尝试从采样和测量的实验过程中尽可能准确地重现函数时，这就是我们自然会做的事情。在这种情况下，一个特别有趣的问题是，通过了解操作符在再现核函数上的行为，进一步确定我们可以对作用在再现核函数空间上的运算符的属性说多少。因此，所提出的程序的更具体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运算符的属性，同时也获得更深入的理解空间本身。 拟议研究计划的科学方法使用了现代和经典数学分析多个领域的方法。除了标准算子理论和复分析技术外，还涉及一般测度论、几何函数论和线性代数方法。 我的团队方法的新颖之处在于，我们仅提取要探索的空间和运算符的一些基本必需属性，从而尝试概括处理特定空间上的特定运算符的几个最新分类结果。我希望与我的 HQP 一起推导出一个模型，一方面可以解决更一般性的问题，另一方面可以更深入地了解探索对象的结构。除了数学之外，这些类型的结果也很有趣，并且可以直接应用于自然科学和工程的其他领域，例如量子力学、量子信息、控制理论、机器学习、图像处理和统计学。