Mathematical Sciences: Analysis in Spaces of Analytic and Harmonic Functions

Axler 9424417研究人员将对复变函数理论和泛函分析之间的相互作用产生的问题进行研究。主要的重点是研究某些重要的解析和调和函数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、超循环性、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调和函数的符号处理软件，适用于纯数学家和应用数学家以及在研究中使用调和函数的科学家/工程师。算符理论起源于数学物理，特别是量子力学的研究。原子的模型化需要发展非对易分析。算子的交换子是非交换性的度量。在势函数的研究中会遇到调和函数，其起源可以追溯到静电学。解析函数是调和函数的复类比。这个项目的很大一部分内容涉及算子及其作用于解析函数空间的交换子的分析。这个项目的研究应该对数学分析界和其他从事非对易模型和势能理论工作的科学家有所裨益。***

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. ***