RUI: Complex Symmetric Operators - Theory and Applications

希尔伯特空间算子的研究联系了各种各样的学科，从控制论，逼近论，量子力学，函数论，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，虽然包括许多众所周知的和有用的类别，但直到最近才得到充分的研究。广义地说，一个希尔伯特空间算子被称为复对称的，如果它有一个关于某个标准正交基的对称矩阵表示（在复数域上）。这个令人惊讶的大类包括所有正常运营商，截断Toeplitz运营商（包括约旦模型运营商和有限Toeplitz矩阵），汉克尔运营商，和许多非正常的积分和微分运营商（包括经典的沃尔泰拉运营商和某些辅助运营商的复杂尺度方法产生的薛定谔运营商）。 PI将在抽象层面上检查这些运算符，同时还考虑与函数论，矩阵分析和其他领域接口的一些具体问题。 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，并赞助本科生研究。新兴的复对称算子理论已经被证明是本科生研究的沃土。 从拟议的项目产生的许多问题是适合本科生的研究和PI将招募来自不同背景的学生对他们的工作。PI还计划带他的本科研究人员参加与该提案相关的会议。 学生们得到的好处是多方面的。例如，他们可以看到数学家在他们的自然元素，了解前沿研究，是相关的自己，与研究人员在社会背景下互动，并坦率地与研究生？这是本科院校所没有的。 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。

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.