凸域上的复合算子及其非交换拓展

Composition operators have been widely studied due to many interesting problems from operator theory and function theory can be modelled into some corresponding problems in the theory of composition operators. This project aims to several key problems of composition operators on some typical convex domains by using the localization theory of function theory and some new methods and tools. That is, we will first describe the topological structure of composition operators on Bergman spaces over the unit disk. We then continue to study the compact difference of composition operators on Hardy spaces over the unit disk, which has been an open problem since it has been raised by Shapiro and Sundberg in 1990. Second, we study the boundedness of high dimensional composition operators over the unit ball, the polydisk and some model domains, which are basic open problems in the theory of composition operators. This will reflect the essential difference between high-dimensional composition operators and one-dimensional composition operators, and establish some relationship between the unit ball and polydisk, which are not holomorphic equivalence. Third, we will study composition operators on the spaces of Dirichlet series over some right-half planes, which will explore some connections between composition operators and Riemann conjecture. Last, we will study composition operators over noncommutative multivariable operator domains, which is one new development direction of operator theory. The research of this project will enrich the content of composition operators and perfect the theory of composition operators. In addition, the research will promote the feedback on function spaces from functional analysis.

复合算子由于可模型化算子理论和函数论中许多深层次的问题而受到广泛研究. 本项目拟结合函数论的局部化理论开采一些新方法, 研究几类典型凸域上复合算子的几个核心问题: 1.刻画单位圆盘上Bergman空间上复合算子的拓扑结构, 进而研究单位圆盘上Hardy空间上复合算子的紧差问题, 该问题自1990年提出之后一直是公开问题; 2.刻画单位球、多圆柱、模型域上高维复合算子的有界性问题, 这是复合算子理论界长期关注的焦点问题, 以此反映高维复合算子与一维复合算子的本质差异性, 建立单位球与多圆柱这两个不全纯等价域的内在联系; 3.深入研究半平面上Dirichlet级数上的复合算子, 探索复合算子与Riemann猜想的某种联系; 4.系统研究非交换域上的复合算子, 这是算子理论界新的发展方向. 本项目的研究必将丰富复合算子的研究内容，完善复合算子理论, 推动泛函分析基本理论对函数论的实质反馈.

复合算子由于可模型化算子理论和函数论中许多深层次的问题而受到广泛研究.本项目根据国内外复合算子及相关领域的最新研究现状与发展动态,以问题驱动为导向,结合函数论的局部化理论开采了一些新方法,研究了几类典型凸域上复合算子的几个核心问题:1.刻画了单位圆盘Bergman空间上复合算子的拓扑结构,完全解决了1990年Shapiro与Sundberg提出的一个公开问题;2.刻画了单位球与多圆柱上高维复合算子的有界性问题,这是复合算子理论界长期关注的焦点问题,反映了高维复合算子与一维复合算子的本质差异性;3.深入研究半平面上Dirichlet级数上的复合算子;4.系统研究非交换域上的复合算子,建立复合算子的非交换理论,开辟了复合算子理论新的发展方向.本项目的研究丰富了复合算子的研究内容,完善了复合算子理论,推动了泛函分析基本理论对函数论的实质反馈.