Mathematical Sciences: Analysis in Spaces of Analytic and Harmonic Functions

研究者将对复函数理论与泛函分析相互作用所产生的问题进行研究。主要的重点将放在研究解析函数和调和函数的某些重要空间以及这些空间上的算子。要考虑的主题包括调和函数的逼近、Berezin变换的迭代、超环性、Koenigs函数、复合算子、Toeplitz算子、Hankel算子和Bergman空间。该项目还将包括开发用于谐波函数符号操作的软件，适合纯数学家和应用数学家以及在其研究中使用谐波函数的科学家/工程师使用。算符理论起源于数学物理学，特别是量子力学的研究。原子的建模需要非交换分析的发展。算子的对易子是非对易性的度量。谐波函数是在研究势函数时遇到的，势函数的起源可以追溯到静电学。解析函数是调和函数的复类似物。这个项目的很大一部分涉及算子及其对解析函数空间上的对易子的分析。这个项目的研究应该对数学分析界和其他涉及非交换模型和势理论的科学家有益。＊＊＊

Axler 9424417 The investigators will conduct research on problems arising from the interaction between complex function theory and functional analysis. Primary emphasis will rest on the study of certain important spaces of analytic and harmonic functions and operators on these spaces. The topics to be considered include approximation by harmonic functions, iterations of the Berezin transform, hypercyclicity, Koenigs functions, composition operators, Toeplitz operators, Hankel operators and Bergman spaces. The project will also include the development of software for the symbolic manipulation of harmonic functions, appropriate for use by pure and applied mathematicians and scientists/engineers who utilize harmonic functions in their research. Operator theory has its origins in mathematical physics, in particular, in the study of quantum mechanics. The modeling of the atom required the development of non-commutative analysis. The commutator of operators is the measure of non-commutativity. Harmonic functions are encountered in the study of potential functions whose origins can be traced to electrostatics. Analytic functions are complex analogues of harmonic functions. A good portion of this project deals with operators and the analysis of their commutators that acting on spaces of analytic functions. The research on this project should be of benefit to the mathematical analysis community and to other scientists whose work involves non-commutative models and potential theory. ***