高维加权Bergman空间上的n-Berezin变换及Toeplitz算子的相似不变量

The commutant of operators and the theory of reducing subspace have the important significance in the reseach of operator structure, especially in the reseach of the problem of invariant subspace. The reseach of similarity invariant of operators on the different spaces is an important branch and full of vitality in operator theory and operator algebra. In solving these problems, many people try to use different kinds of methods and techniques in mathematics. The commutant of an analytic Toeplitz operator on the Hardy space has been studied extensively in the literature. In recent years, some mathematicians begin to reseach the properties of analytic Toeplitz operators on the Bergman space, a few of scholars begin to reseach the corresponding properties of analytic Toeplitz operators on the high dimensional weighted Bergman space. The reseach contents in this project include the followings:(1)Find and summarize some integral and inner product representations of n-Berezin transform on the high dimensional weighted Bergman space.(2)Characterize the commutant of a class of analytic Toeplitz operators on the high dimensional weighted Bergman space.(3)Research the strongly irreducibility of a class of analytic Toeplitz operators on the high dimensional weighted Bergman space.(4)Search for the similarity invariant of a class of analytic Toeplitz operators on the high dimensional weighted Bergman space.

算子的换位和约化子空间理论对算子的结构研究，特别是不变子空间问题具有重要意义，而运用各种数学方法和数学工具寻找不同空间上算子的相似不变量更是算子理论及算子代数中重要且富有生命力的研究课题。Hardy空间上算子的换位自七十年代以来曾被国内外许多学者深入地研究过。近些年来不少学者开始对Bergman空间上解析Toeplitz算子的性质进行研究，而对于高维加权Bergman空间上解析Toeplitz算子相关性质的研究却不多见。本项目计划围绕四方面内容进行研究：(1)给出高维加权Bergman空间上n-Berezin变换的内积表示和积分表示；(2)刻画高维加权Bergman空间上一类解析Toeplitz算子的换位；(3)研究高维加权Bergman空间上解析Toeplitz算子的强不可约性；(4)寻找高维加权Bergman空间上解析Toeplitz算子的相似不变量。

算子的换位和约化子空间理论对算子的结构研究，特别是不变子空间问题具有重要意义，而运用各种数学方法和数学工具寻找不同空间上算子的相似不变量更是算子理论及算子代数中重要且富有生命力的研究课题。Hardy空间上算子的换位自七十年代以来曾被国内外许多学者深入地研究过。近些年来不少学者开始对Bergman空间上解析Toeplitz算子的性质进行研究，而对于高维加权Bergman空间上解析Toeplitz算子相关性质的研究却不多见。本项目完成以下五方面内容：(1)刻画了Fock空间上乘法算子的相似性及约化子空间问题；(2)刻画了Sobolev圆盘代数上乘法算子的相似性及约化子空间问题；(3)研究了单位球加权Bergman空间上解析Toeplitz算子的拟仿射性；(4)研究了双圆盘加权Bergman空间上乘法算子的相似性及约化子空间问题；(5)刻画了n-移位加带特定权的乘法算子的相似性及约化子空间问题。本项目共发表主要论文7篇，对于丰富函数空间上算子理论具有重要的理论意义。