Complex symmetric operators and function theory

私家侦探将研究复对称算子，一类希尔伯特空间算子，虽然包括许多众所周知的和有用的类，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（包括经典的沃尔泰拉算子和由Schrodinger算子的复尺度方法产生的某些辅助算子）。和合作者包括一个一般的结构定理复杂的对称运营商和新的变分原理的奇异值的紧凑复杂的对称运营商，表明发挥了一定的反线性对称的基本作用。 拟议的项目旨在继续对复对称算子（或其某些子类）进行一般性研究，同时也侧重于将现有成果应用于相关领域（例如函数论和矩阵论）。 为了扩大这项工作的影响和适用性，P.I.他将与数学、物理和工程领域的同事合作，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之而来的结果将是有意义的算子理论，函数理论，矩阵理论，和一些特定的分支数学物理和工程的当前研究。

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.