Analysis of Toeplitz Operators

Toeplitz算子分析

• 批准号: 0200607
• 负责人: Dechao Zheng
• 金额: $ 10.41万
• 依托单位: Vanderbilt University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-06-01 至 2006-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0200607&HistoricalAwards=false
• 关键词: Analysis; Toeplitz; Operators

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.

[摘要]研究功能理论与功能分析相互作用中产生的问题。重点将放在一些函数空间上Toeplitz算子的研究上。要考虑的主题包括Hardy空间或Bergman空间上“与函数理论密切相关”的紧算子、有界Toeplitz算子、对偶Toeplitz算子的代数和谱性质、Segal-Bargmann空间上的Hankel和Toeplitz算子以及Segal-Bargmann空间的拟不变子空间。这个项目的中心问题是建立这些算子的基本性质和它们的符号的解析和几何性质之间的关系。所提出的工作涉及算子理论和复杂分析的思想和问题。算符理论起源于物理学中用于研究某些偏微分方程的思想，随着量子力学的出现而变得越来越重要。人们对Toeplitz运营商的兴趣持续增长，至少有两个原因。除了微分算子之外，Toeplitz算子也是非自伴随算子中最重要的一类，它们是算子理论、函数理论、调和分析和算子代数之间富有成果的相互作用的一个迷人的例子。另一方面，Toeplitz算子在物理学、概率论、信息论和控制论以及其他几个领域的各种问题中都很重要。