RUI: Operators on Hilbert space

RUI：希尔伯特空间上的算子

• 批准号: 1265973
• 负责人: Stephan Garcia
• 金额: $ 19.9万
• 依托单位: Pomona College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-08-01 至 2017-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1265973&HistoricalAwards=false
• 关键词: RUI; Operators; Hilbert; space

This project will study complex symmetric operators and truncated Toeplitz operators, aiming to clarify the intriguing relationship between these two interrelated classes of Hilbert space operators. Complex symmetric operators are a broad class of operators that has not been adequately studied in generality until recently. The study of truncated Toeplitz operators, a rapidly growing branch of function-theoretic operator theory, has undergone spirited development stemming from a seminal 2007 paper of Sarason. A recent series of articles by the principal investigator and his growing list of collaborators have unearthed surprising links between these two classes. Research on this subject has the potential to be transformative, having relevance to a number of fields. For instance, connections to function theory and matrix analysis have already engaged researchers from both large institutions and small colleges. Moreover, the study of complex symmetric and truncated Toeplitz operators has already proven to be fertile ground for undergraduate research.The study of linear operators on Hilbert space has its modern roots in the seminal work of von Neumann, who in the early twentieth century developed a rigorous framework in which to describe quantum mechanics. Since then, operator theory has found applications in electrical engineering, quantum physics, image processing, and statistics, to name a few areas outside of pure mathematics. This project will study two novel, but surprisingly ubiquitous, classes of operators, while also considering several specific questions that interface with other researchers in the mathematical sciences. To do this, the principal investigator will collaborate with colleagues old and new, as well as sponsor undergraduate research. Indeed, many questions stemming from this project are suitable for undergraduate research and the principal investigator will recruit a diverse array of students to work on them. These students will be equipped with the skills, expertise, confidence, and passion necessary to pursue careers in the mathematical sciences. This project will create not only new mathematics, but also the new mathematicians necessary to enhance and enrich the nation's infrastructure for research and education.

本专题将研究复对称算子和截断Toeplitz算子，旨在阐明这两类相互关联的希尔伯特空间算子之间的有趣关系。 复对称算子是一类广泛的算子，直到最近才得到充分的研究。 截断Toeplitz算子的研究是函数论算子理论的一个快速发展的分支，从2007年Sarason的一篇开创性论文开始，已经经历了蓬勃的发展。 首席研究员和他越来越多的合作者最近发表的一系列文章揭示了这两个类别之间令人惊讶的联系。 关于这一主题的研究有可能产生变革，与若干领域有关。 例如，与函数理论和矩阵分析的联系已经吸引了来自大型机构和小型学院的研究人员。 此外，研究复对称和截断的Toeplitz算子已经被证明是大学生研究的肥沃土壤，希尔伯特空间上的线性算子的研究在冯·诺依曼（von Neumann）的开创性工作中有其现代根源，他在世纪早期建立了一个严格的框架来描述量子力学。 从那时起，算子理论已经在电气工程，量子物理，图像处理和统计学中找到了应用，仅举几个纯数学之外的领域。 该项目将研究两种新颖的，但令人惊讶的是无处不在的运算符类，同时还考虑与数学科学中的其他研究人员接口的几个具体问题。 为此，首席研究员将与新老同事合作，并赞助本科生研究。 事实上，这个项目产生的许多问题都适合本科生研究，首席研究员将招募各种各样的学生来研究它们。 这些学生将配备必要的技能，专业知识，信心和激情，以追求职业生涯中的数学科学。 该项目不仅将创造新的数学，而且还将创造新的数学家，以加强和丰富国家的研究和教育基础设施。