Operators on reproducing kernel Banach spaces of analytic functions

拟议的研究方案是在数学分析领域。它的总体目标是增强算子理论和复数分析方面的知识，以及它们之间的相互作用。这些领域与几门自然科学和工程学有着密切的联系，在大多数情况下，问题是由特定的应用直接引起的。*虽然函数是经典分析的主要探索对象，但现代分析的目标是探索函数类的变换。在这个方案中，我们计划研究的函数类是解析函数的再生核Banach和Hilbert空间，而变换是由这些空间的结构自然决定的线性算子。这类空间的一些著名和广泛探索的例子是Bergman、Hardy、Dirichlet、Bloch和Besov空间。所讨论的算子类包括乘法算子、复合算子、Toeplitz算子、积分算子和条件期望算子。*这些类型的空间的一个很好的性质是我们可以使用再生核来计算空间中的函数在其定义域的特定点上的值。当我们试图尽可能准确地从采样和测量的实验过程中复制函数时，这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，通过知道算子如何作用于再生核函数，进一步确定我们可以对作用于再生核函数空间的算子的性质说多少。因此，建议的方案的更具体的目标是分类和确定一大类算子的性质，通过反映它们作用于的再生核函数空间的结构，同时也加深对空间本身的理解。*建议的研究方案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。除了标准的算子论和复数分析技术外，它还涉及一般测度论、几何函数论和线性代数方法。*我的团队方法的新奇之处在于，我们只提取了要探索的空间和算子的几个基本必需性质，从而试图推广几个较新的分类结果，这些结果涉及特定空间上的特定算子。与我的HQP一起，我希望得出一个模型，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层次的洞察。除了数学之外，这些类型的结果还令人感兴趣，并直接应用于自然科学和工程的其他领域，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。

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.