RUI: Complex Symmetric Operators - Theory and Applications

希尔伯特空间算子的研究涉及广泛的学科，从控制论、逼近论、量子力学、函数论、信号处理、非对易几何和矩阵理论，仅举几例。PI将研究复对称算子，这是一类广泛的希尔伯特空间算子，虽然包含了许多著名和有用的类，但直到最近才被充分研究。松散地说，如果一个Hilbert空间算子关于某一正交基具有对称的矩阵表示(在复数域上)，则称它为复对称的。这类令人惊讶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算子和许多非正规积分和微分算子(包括经典的Volterra算子和由薛定谔算子的复标度法产生的某些辅助算子)。PI将在抽象级别上研究这些运算符，同时还会考虑与函数论、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，并资助本科生的研究。新兴的复对称算子理论已经为本科生的研究提供了肥沃的土壤。该项目提出的许多问题都适合本科生研究，PI将招募来自不同背景的学生来研究这些问题。国际和平研究所还计划带他的本科生研究人员参加与这项提议相关的会议。给学生们带来的好处很多。例如，他们可以看到数学家的自然元素，了解与他们自己相关的尖端研究，在社会背景下与研究人员互动，并与研究生坦率交谈？本科生学校里没有的东西。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/或演讲。

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.