Operator Theory and Complex Analysis

算子理论与复分析

• 批准号: 9970376
• 负责人: Nathan Feldman
• 金额: $ 5.25万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-06-01 至 1999-10-05
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9970376&HistoricalAwards=false
• 关键词: Operator; Theory; Complex; Analysis

Proposal: DMS-9970376Principal Investigator: Nathan S. FeldmanAbstract: Feldman will conduct research in operator theory and its interplay with complex analysis. Primary emphasis will be placed on studying special classes of linear operators that are close to normal operators, such as subnormal, hyponormal, and essentially subnormal operators. Questions regarding various forms of cyclicity for cosubnormal and cohyponormal operators will be investigated, as will the relationship between cyclicity and the existence of intertwining maps. Other topics to be explored include: the existence of hyperinvariant subspaces for subnormal operators; "generalized eigenvectors" for adjoints of subnormal operators; the principal function for a subnormal operator, and its relations with such classes of operators as Toeplitz operators, Hankel operators, and self-commutators.Operator theory is a natural extension of the ideas of matrices and linear algebra to infinite dimensions. In simple terms, an operator is just an infinite array or matrix of numbers and is used in linear problems involving infinitely many unknown variables. Although many intractable real world problems are not linear, one can frequently "linearize" a given problem to produce a new problem that is both linear and solvable and that has a solution which closely approximates -- hence provides valuable information about -- the solution of the original physical problem. Perhaps the simplest example of this is the idea of the derivative from calculus, which is used to approximate a curve by a straight line (something that is linear). In a multivariable setting, the precise analogue of the derivative is a linear operator. Recognizing how useful and powerful a tool the derivative of even a single variable function has become in all areas of science, one can readily appreciate how important it is to understand fully the structure of linear operators. While some operators can be rather "pathological" in character, the subnormal operators that are the objects of study in this project have a great deal of structure: they are very natural themselves, and they arise in natural ways. Operator theory and linear algebra have had a profound impact on all branches of mathematics, physics, and engineering. It is hoped that Feldman's research will likewise exert an influence not only on pure mathematics but on the applied sciences as well.

摘要：Feldman将研究算子理论及其与复分析的相互作用。主要的重点将放在研究与正规算子接近的特殊类别的线性算子，如次正规、次正规和本质上的次正规算子。本文将研究共次正规和共次正规算子的各种形式的环性问题，以及环性与交织映射存在性之间的关系。其他要探讨的主题包括：次正规算子的超不变子空间的存在性；次正规算子伴随的“广义特征向量”次正规算子的主函数，以及它与Toeplitz算子、Hankel算子和自对易子算子的关系。算子理论是矩阵和线性代数思想向无限维的自然延伸。简单地说，一个算子就是一个无限的数字数组或矩阵，用于涉及无限多个未知变量的线性问题。尽管现实世界中许多棘手的问题都不是线性的，但人们经常可以将一个给定的问题“线性化”，从而产生一个新的问题，这个问题既是线性的，又是可解的，而且它的解与原始物理问题的解非常接近，从而提供了有关原始物理问题解的有价值的信息。也许最简单的例子是微积分中的导数，它被用来用一条直线（线性的东西）近似一条曲线。在多变量设置中，导数的精确类比是线性算子。认识到即使是单一变量函数的导数在所有科学领域中是多么有用和强大的工具，人们很容易意识到充分理解线性算子的结构是多么重要。虽然有些操作符在性质上可能相当“病态”，但作为本项目研究对象的次正常操作符具有大量结构：它们本身非常自然，并且以自然的方式出现。算子理论和线性代数对数学、物理和工程的所有分支都产生了深远的影响。人们希望费尔德曼的研究不仅对纯数学，而且对应用科学也会产生同样的影响。