乘法算子，Hankel算子，Toeplitz算子及Toeplitz代数

The main objective of this project is to deepen the PI's previous investigations into behavior of classical Hankel operators, Toeplitz operators and multiplication operators on the Bergman space and the Hardy space via harmonic analysis and complex analysis...1. One goal of this project is to use some ideas and methods in harmonic analysis to study Hankel operators on the Hardy space of the bidisk...2. Another goal of this project is to study Toeplitz operators on the Bergman space and their spectrum by using some ideas in harmonic analysis and complex analysis. Sheldon Axler and the PI proved that on the Bergman space, compact Toeplitz oper- ators are completely determined by its Berezin transform. Carl Sundberg and the PI answer an open question on the essential spectrum of Toeplitz operators since 1980s...3. Another goal is to develop operator theory on the Bergman space via the Hardy space of the bidisk...4. Using complex analysis, R. Douglas, S. Sun and the PI show that the commutant of a multiplication operator by a finite Blaschke product and the adjoint.of the operator has dimension equal to the number of connected components of Rie- mann surface associated with the Blashcke product. The PI intends to study the multiplication operator by using complex analysis, complex geometry, Riemann surfaces, complex differential equations...5. The famous Chang-Marshall Theorem gives a complete characterization of Douglas algebras. S. Axler and the PI established the analogous Chang-Marshall theorem for disk algebras. Another goal is to study the structure of Quantum Douglas algebra, which are a family of Toeplitz algebras on Bergman spaces...The PI is advising undergraduate students and graduate students at Chongqing University and always encourages them to choose scientific careers. Ideas and techniques from functional analysis and operator theory are powerful tools in studying matrix theory arising from engineering linear systems problems, like those arising in the design and control of automatic controllers. In particular, Hankel operators and Toeplitz operators are of importance in applied mathematics such as system theory, and stationary stochastic processes.

本项目的主要目标是研究伯格曼空间和哈代空间上的经典汉克算子、托普利兹算子和乘法算子。本项目的一个目的是应用调和分析的观点和方法来研究双圆盘上哈代空间的汉克算子。第二个目的是用调和分析和复分析的一些观点研究伯格曼空间上得托普利兹算子和他们的谱。另一个目的是通过双圆盘的哈代空间来发展伯格曼空间上的算子理论。PI希望用复分析、复几何、黎曼曲面的办法来研究乘法算子. 著名的Chang-Marshall定理给出道格拉斯代数的完全刻画。S.Axler和PI下在圆盘上建立了类似的Chang-Marshall定理。另一个目的就是研究量子道格拉斯代数（Quantum Douglas Algebra）。来自于泛函分析和算子代数的观点和方法是研究从工程线性问题（比如规划和自动控制）产生的矩阵理论的有力工具。特别的，汉克算子和托普利兹算子在诸如系统理论、静态随机过程等应用数学中也有重要的作用.

本项目主要研究乘法算子、Hankel算子、Toeplitz算子以及Toeplitz代数。 我们的具体研究内容是：考虑Bergman 空间上Toeplitz算子的有界性、正定性、可逆性，谱结构，以及模空间上截断Toeplitz算子的紧性，解析Toeplitz算子的酉等价、约化子空间等问题。我们的目标是综合调和分析、复分析、几何分析和函数论的方法深入研究Bergman空间上的Toeplitz算子、Hankel算子以及乘法算子的经典性质。经过四年的不懈努力， 我们给出了一些使得Bergman Toeplitz可逆的充分条件，而且研究了有界调和符号的Bergman Toeplitz算子的可逆性，完全刻画了非负有界符号的Toeplitz算子的可逆性；我们完全刻画了模空间上的截断Toeplitz算子的紧性、完全建立了Bergman空间上Toeplitz行列式的第一Szego定理；清晰的刻画了在何种条件下，Bergman空间上两个调和符号的Toeplitz算子的乘积等于某个Toeplitz算子与一个有限秩算子之和；此外，我们还考虑了高维情形下的乘法算子的约化子空间以及相关的von Neumann代数等问题。