RUI: Operators on Hilbert space

本项目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Hilbert空间算子之间的有趣关系。复对称算子是一类广义的算子，直到最近才得到充分的一般性研究。截断Toeplitz算子的研究是函数算子理论中一个快速发展的分支，自2007年Sarason的一篇开创性论文以来，得到了蓬勃的发展。最近，这位首席研究员和他越来越多的合作者发表了一系列文章，揭示了这两个阶层之间令人惊讶的联系。对这一主题的研究可能具有变革性，与许多领域相关。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，复杂对称和截断Toeplitz算子的研究已经被证明是本科生研究的沃土。希尔伯特空间上线性算子的研究在冯·诺伊曼的开创性工作中有其现代根源，他在20世纪初发展了一个严格的框架来描述量子力学。从那时起，算符理论在电子工程、量子物理、图像处理和统计学等纯数学以外的领域得到了应用。该项目将研究两种新颖但令人惊讶的普遍存在的算子类，同时也考虑与其他数学科学研究人员交互的几个具体问题。为此，首席研究员将与新老同事合作，并资助本科生研究。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这些问题。这些学生将具备从事数学科学事业所必需的技能、专业知识、信心和热情。这个项目不仅将创造新的数学，而且还将培养新的数学家，以加强和丰富国家的研究和教育基础设施。

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.