Analysis of Toeplitz Operators

派：德超郑命题编号：0200607研究函数论与泛函分析相互作用产生的问题。主要的重点是研究某些函数空间上的Toeplitz算子。要考虑的主题包括Hardy空间或Bergman空间上与函数论密切相关的紧算子，有界Toeplitz算子，对偶Toeplitz算子的代数性质和谱性质，Segal-Bargmann空间上的Hankel和Toeplitz算子，以及Segal-Bargmann空间的拟不变子空间。这个项目集中于建立这些运算符的基本性质与其符号的解析和几何性质之间的关系的中心问题。所提出的工作涉及到算子理论和复变分析的思想和问题。算符理论起源于用于研究物理学中某些偏微分方程的思想，并随着量子力学的出现而变得越来越重要。人们对Toeplitz算子的兴趣持续增长至少有两个原因。除微分算子外，Toeplitz算子还是一类重要的非自伴算子，它们是算子理论、函数论、调和分析和算子代数等学科相互作用的一个有趣例子。另一方面，Toeplitz算子在物理、概率论、信息和控制理论以及其他几个领域的各种问题中都具有重要意义。

PI: Dechao ZhengProposal Number: 0200607 ABSTRACTResearch will be conducted on problems arising from the interaction between function theory and functional analysis. Primary emphasis will rest on the study of Toeplitz operators on some function spaces. The topics to be considered include compact operators ``closely associated with function theory'' on the Hardy space or the Bergman space, bounded Toeplitz operators, algebraic and spectral properties of dual Toeplitz operators, Hankel and Toeplitz operators on the Segal-Bargmann spaces and quasi-invariant subspaces of the Segal-Bargmann spaces. This project focuses on the central problem of establishing the relationship between the fundamental properties of those operators and analytic and geometric properties of their symbols. The proposed work involves ideas and problems from operator theory and complex analysis. Operator Theory grew out of ideas used to study certain partial differential equations arising in physics, and became increasinglyimportant with the advent of Quantum Mechanics. There are at least two reasons for the continuous and increasing interest in Toeplitz operators. In addition to differential operators, Toeplitz operators constitute one of the most important classes of non-selfadjoint operators and they are a fascinating example of the fruitful interplay between such topics as operator theory, function theory, harmonic analysis and operator algebras. On the other hand, Toeplitz operators are of importance in connection with a variety of problems in physics, probability theory, information and control theory, and several other fields.