RUI: Complex Symmetric Operators - Theory and Applications

RUI：复杂对称运算符 - 理论与应用

• 批准号: 1001614
• 负责人: Stephan Garcia
• 金额: $ 16.49万
• 依托单位: Pomona College
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-07-01 至 2014-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1001614&HistoricalAwards=false
• 关键词: RUI; Complex; Symmetric; Operators; Theory

The study of Hilbert space operators links a wide variety of disciplines, ranging from control theory, approximation theory, quantum mechanics, function theory, signal processing, noncommutative geometry, and matrix theory, to name but a few. The PI will study complex symmetric operators, a broad class of Hilbert space operators which, while encompassing many of the well-known and useful classes, has not been adequately studied in generality until recently. Loosely put, a Hilbert space operator is called complex symmetric if it has a symmetric matrix representation (over the complex field) with respect to some orthonormal basis. This surprisingly large class includes all normal operators, truncated Toeplitz operators (including Jordan model operators and finite Toeplitz matrices), Hankel operators, and many non-normal integral and differential operators (including the classical Volterra operator and certain auxiliary operators produced by the complex scaling method for Schrodinger operators). The PI will examine these operators at the abstract level while also considering a number of specific questions that interface with function theory, matrix analysis, and other areas. For instance, connections to complex analysis have already engaged a number of researchers from both large institutions and small colleges. The PI will collaborate with colleagues old and new, as well as sponsor undergraduate research.The emerging theory of complex symmetric operators has already proven fertile ground for undergraduate research. Many questions stemming from the proposed project are suitable for undergraduate research and the PI will recruit students from diverse backgrounds to work on them. The PI also plans to take his undergraduate researchers to conferences related to this proposal. The benefits for the students are many. For instance, they get to see mathematicians in their natural element, learn about cutting-edge research that is relevant to their own, interact with researchers in a social context, and speak candidly with graduate students ? something not available at an undergraduate institution. As part of their sustained research experience, student researchers will present posters and/or give talks in venues appropriate for undergraduate research.

希尔伯特空间算符的研究涉及广泛的学科，从控制理论、近似理论、量子力学、函数理论、信号处理、非交换几何和矩阵理论，仅举几例。PI将研究复对称算子，这是希尔伯特空间算子的一个广泛的类别，虽然包含了许多众所周知和有用的类别，但直到最近才得到充分的一般研究。宽泛地说，如果一个希尔伯特空间算子（在复域上）对某些标准正交基具有对称的矩阵表示，那么它就被称为复对称算子。这个惊人的大类包括所有正规算子，截断Toeplitz算子（包括Jordan模型算子和有限Toeplitz矩阵），Hankel算子，以及许多非正规积分和微分算子（包括经典Volterra算子和某些由薛定谔算子的复标度法产生的辅助算子）。PI将在抽象层面检查这些算子，同时也考虑与函数理论、矩阵分析和其他领域相关的一些具体问题。例如，与复杂分析的联系已经吸引了来自大型机构和小型学院的许多研究人员。PI将与新老同事合作，并资助本科生研究。新兴的复对称算子理论已经被证明是大学生研究的沃土。项目提出的许多问题都适合本科生进行研究，项目负责人将招募来自不同背景的学生进行研究。PI还计划带他的本科生研究人员参加与该提案相关的会议。bene& # 64257;给学生的t是很多的。例如，他们可以看到数学家在他们的自然元素中，了解与他们自己相关的前沿研究，在社会背景下与研究人员互动，并与研究生坦诚交谈。本科院校没有的东西。作为他们持续研究经验的一部分，学生研究人员将在适合本科生研究的场所展示海报和/或发表演讲。