多重调和Bergman空间上Toeplitz算子的代数性质的研究

Operator theory on function space is a hot problem for discussion in the field of both analyses and operator theory, which establishes the close relation between function space and operator theory..The functions in harmonic Bergman space and Hardy space of unit circle have similar structures, that is they can be considered as the sum of holomorphic function and antiholomorphic function. So we pay attention to whether the results in harmonic Bergman space are analogous to that in Hardy space, and put many new questions. In contrast with operator theroy in one variable, operator theory in several variables is more complicated, and there are much more interesting questions to be resolved..In this project, we will study the algebraic peoperties of the Toeplitz operators on pluriharmonic Bergman space. By Mellin transform, Berezin transform, multiple Fouriers series theory, vector-valued operator theory, several complex variables function theory and complex geometry, some problems about algebraic properties of Toeplitz operators on pluriharmonic Bergman space, such as the product, (semi)commutativity and essential commutativity, will be studied.

因为建立了函数空间和算子理论之间的紧密联系，函数空间上算子理论的研究一直是分析领域和算子理论领域的热门研究对象。.调和Bergman空间中的函数与单位圆周Hardy空间中的函数有着相似的结构（即都可看作解析函数与余解析函数的和）。因此人们关注调和Bergman空间上的算子理论是否有类似Hardy空间上算子理论中的结果，并衍生了许多新的问题。多变量函数空间上算子理论比单变量函数空间上算子理论复杂得多，问题也更丰富。.本项目致力于多重调和Bergman空间上Toeplitz算子代数性质的研究。以Mellin变换、Berezin变换、多重 Fourier级数理论、向量值算子理论、多复变函数理论和复几何理论为工具，研究多重调和Bergman空间上Toeplitz算子的乘积、（半）交换性和本性交换性等代数性质问题。

函数空间上算子理论的研究一直是分析领域和算子理论领域的热门研究对象。调和Bergman空间中的函数与单位圆周Hardy空间中的函数有着相似的结构（即都可看作解析函数与余解析函数的和）。因此人们关注调和Bergman空间上的算子理论是否有类似Hardy空间上算子理论中的结果，并衍生了许多新的问题。多变量函数空间上算子理论比单变量函数空间上算子理论复杂得多。.本项目研究了多重调和Bergman空间上Toeplitz算子的代数性质。给出了单位球的Bergman空间上以径向函数和多重调和函数为符号Toeplitz算子的半交换和交换的充要条件。概括总结了赋范空间上的Aleksandrov–Benz–Rassias问题。