Fock空间上的若干问题研究

Functional analysis and complex analysis are very important research areas in the basic mathematical theory, and function space theory is a very basic and important research direction in the field of modern analysis. Fock spaces in quantum physics, Harmonic Analysis on the Heisenberg Groups and Partial Differential Equations, etc. are widely used, so researching a number of issues in the Fock space has important scientific significance. ..This project will use the Fock space as a platform: researching norm convergence of Taylor series of a function in the Fock space; differences of zero sequences in different Fock spaces; constructing a proper weight function and characterizing the compactness of operators in the Toeplitz algebra of the weighted Fock space induced by the weight function; the existence of finite rank Toeplitz operators and zero products of Toeplitz operators，etc. ..The research of this project will involve into the function theory, Banach algebras, topology, harmonic analysis and other research areas, construct a intrinsic relation between the properties of Toeplitz operators and the geometric properties of their symbols, give full applications to the function theory and Banach algebra techniques in operator theory, so this will have a positive effect and influence on the research of this field and related areas.

泛函分析与复分析是基础数学理论中非常重要的研究领域，而函数空间理论又是现代分析领域中非常基础而又重要的研究方向。Fock空间在量子物理、Heisenberg群上的调和分析和偏微分方程等方面都有着广泛的应用，从而对Fock空间中的若干问题的研究具有非常重要的科学意义。本项目以Fock空间为平台，研究Fock 空间中函数的泰勒级数在范数拓扑下的收敛性；不同Fock空间的零系列的差异性；构造适当的权函数并研究以此权函数诱导的加权Fock 空间的Toeplitz 代数中算子的紧性特征；有限秩Toeplitz算子的存在性与Toeplitz算子的零积问题等等。本项目的研究将涉及到函数论、Banach代数、拓扑、调和分析等研究领域，建立Toeplitz算子的性质与符号函数的几何性质之间的内在联系，充分发挥函数论与Banach代数技巧在算子理论中的应用，这将对本领域和相关领域的研究产生积极的作用和影响。

本项目综合运用函数论、几何、代数、拓扑来研究各种解析函数空间（如Fock空间、Zygmund-型空间等）中的分析性质、空间的结构和不同函数空间之间的算子理论. 本项目所涉及的研究问题是函数空间理论、多复变函数论与算子理论研究领域的现代数学热点课题. 经过三年的努力, 项目按计划圆满完成, 取得了预期的成果. 发表标注本项目资助编号的SCI收录的学术论文16篇, 录用SCI学术论文1篇, 在科学出版社出版学术专著一部.