Complex symmetric operators and function theory

P.I.将研究复对称算子，这是希尔伯特空间算子的一类，虽然包含了许多众所周知和有用的类，但直到最近才得到充分的一般性研究。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、压缩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Volterra算子和某些由薛定谔算子的复标度法产生的辅助算子）。P.I.和合作者最近的成果包括复对称算子的一般结构定理和紧复对称算子的奇异值的新变分原理，这些原理表明了某些反线性对称性所起的基本作用。该项目旨在继续复杂对称算子（或其某些子类）的一般研究，同时也关注现有结果在相关领域的应用（例如函数理论和矩阵理论）。为了扩大这项工作的影响和适用性，私家侦探将与他在数学、物理和工程方面的同事合作，并赞助与该项目相关的本科生研究。希望这些共同的努力和随后的结果将对当前算子理论、函数理论、矩阵理论以及数学物理和工程的一些特定分支的研究具有重要意义。

The P.I. will study complex symmetric operators, a class of Hilbert space operators that, 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, compressed 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).Recent results of the P.I. and collaborators include a general structure theorem for complex symmetric operators and new variational principles for singular values of compact complex symmetric operators that indicate the fundamental role played by certain antilinear symmetries. The proposed project aims to continue the general study of complex symmetric operators (or of certain subclasses thereof) while also focusing on the applications of existing results to related areas (for instance function theory and matrix theory). To broaden the impact and applicability of this work, the P.I. will collaborate with his colleagues in mathematics, physics, and engineering as well as sponsor undergraduate research related to the project. It is hoped that these combined efforts and ensuing results will be significant for current studies in operator theory, function theory, matrix theory, and some specific branches of mathematical physics and engineering.