Analysis of Toeplitz Operators

主要研究者：郑德超提案编号：0200607摘要研究将进行的问题之间的相互作用的功能理论和功能分析。主要的重点将放在研究Toeplitz算子的一些功能空间。 要考虑的主题包括紧凑的运营商"密切相关的功能理论“的哈代空间或伯格曼空间，有界Toeplitz运营商，代数和频谱性质的对偶Toeplitz运营商，汉克尔和Toeplitz运营商的Segal-巴格曼空间和准不变子空间的Segal-巴格曼空间。该项目的重点是建立这些运营商的基本属性和他们的符号的分析和几何属性之间的关系的中心问题。所提出的工作涉及的思想和问题，从算子理论和复分析。算符理论产生于研究物理学中某些偏微分方程的思想，随着量子力学的出现变得越来越重要。Toeplitz算子的持续和不断增长的兴趣至少有两个原因。除了微分算子，Toeplitz算子构成了最重要的一类非自伴算子，他们是一个迷人的例子之间富有成效的相互作用等主题的算子理论，函数理论，调和分析和算子代数。另一方面，Toeplitz算子在物理学、概率论、信息和控制论以及其他几个领域中的各种问题中都很重要。

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.