Multivariable Operator Theory

本研究涉及多变量算子理论，重点关注三个方面：(i)截断矩问题（TMP）表示测度的存在性、唯一性和局部性支持的代数条件；（ii） Reinhardt域上的算子理论，特别强调多变量加权位移的谱和结构性质；（iii）块Toeplitz算子研究中的多变量技术。关于第一个领域，计划是扩展最近关于正矩矩阵的平面扩展和极矩问题的工作（与L. Fialkow， H.M. Möller和S. Yoo联合），这导致了TMP研究的一般框架。首席研究员和他的合作者将开发新的方法和技术，并将其应用于与有限代数变量相关的立方柱关系的情况，以及递归确定矩矩阵的TMPs。第二部分讨论Reinhardt域上泛函Hilbert空间上的乘法算子。该项目的这一部分将通过应用先前的结果（与S.H. Lee和J. Yoon联合）以及采用在首席研究员与P. Muhly和K. Yan的工作中开发的类群技术，将亚正规2变量加权位移的光谱图像研究扩展到次正规。第三个领域处理希尔伯特空间算子的次正态的多变量方法，特别强调标量和块Toeplitz算子。方法是描述2-次异常，然后是k-次异常，最后是次正态。首席研究员将与I.B. Jung， S.H. Lee， W.Y. Lee， S.S. Park和M. Putinar一起进一步发展他之前工作中的想法。作为一个试验场，他将寻找2-次正规算子的模型理论，这一主题将与J. Agler的抽象模型理论建立有用的联系。使用泛函理论和多变量算子理论技术，首席研究员最近证明了块Toeplitz算子的Abrahamse定理的一个版本（与I.S. Hwang和W.Y. Lee共同）。他将寻求这个定理的一个更一般的版本，并解决一个相关的次正规Toeplitz补全问题。物理、数学和工程中的许多问题都可以通过将复杂的物理实体表示为称为矩阵的大型数字和数学符号数组来最好地描述。矩阵帮助我们可视化线性变换如何作用于向量空间；确定它们的结构揭示了转换的重要性质。希尔伯特空间算子是矩阵的无限维推广。向量的泛化通常是一个函数，因此，运算符经常被建模为函数空间上的乘法。该项目的一部分内容是为运营商寻找这样的模型。一旦获得了模型，许多关于算子的基本结构问题就变得自然了。从20世纪50年代开始，亚正规算子的研究取得了很大的成功，其理论对泛函分析、量子力学和工程等领域做出了重要贡献。类似地，标量和块Toeplitz算子类出现在数学和物理的各个领域。研究的另一部分涉及统计学、光谱分析、地球物理学、图像识别、全球定位工具、信号检测理论和经济学中自然发生的逆问题。他在截断矩问题上的研究成果已被应用于最优化理论、实代数几何、数值分析、半定规划和传感器网络定位。该项目旨在解决多变量算子理论中的一些突出问题，同时为女性和少数民族在数学和其他STEM领域的职业生涯创造招聘和保留机会。这个项目中的几个问题是为了产生本科生和研究生可以访问的研究问题而写的。

This research deals with multivariable operator theory, focusing attention on three areas: (i) algebraic conditions for existence, uniqueness, and localization of the support of representing measures for truncated moment problems (TMP); (ii) operator theory over Reinhardt domains, with special emphasis on spectral and structural properties of multivariable weighted shifts; and (iii) multivariable techniques in the study of block Toeplitz operators. Concerning the first area, the plan is to extend recent work on flat extensions of positive moment matrices and extremal moment problems (joint with L. Fialkow, H.M. Möller and S. Yoo), which has led to a general framework for the study of TMP. The principal investigator and his collaborators will develop new methods and techniques, and apply them to the case of cubic column relations associated with finite algebraic varieties, and to TMPs with recursively determinate moment matrices. The second area deals with multiplication operators on functional Hilbert spaces over Reinhardt domains. This part of the project will extend the study of the spectral picture of subnormal 2-variable weighted shifts to hyponormal ones, by applying previous results (joint with S.H. Lee and J. Yoon) and by employing the groupoid techniques developed in the principal investigator's work with P. Muhly and K. Yan. The third area deals with a multivariable approach to subnormality of Hilbert space operators, with special emphasis on scalar and block Toeplitz operators. The approach is to characterize 2-hyponormality, then k-hyponormality, and eventually subnormality. The principal investigator will develop further the ideas in his previous work with I.B. Jung, S.H. Lee, W.Y. Lee, S.S. Park, and M. Putinar. As a testing ground, he will search for a model theory for 2-hyponormal operators, a topic that leads to useful connections with J. Agler's abstract model theory. Using function-theoretic and multivariable operator theory techniques, the principal investigator has recently proved a version of Abrahamse's theorem for block Toeplitz operators (jointly with I.S. Hwang and W.Y. Lee). He will seek a more general version of this theorem, and the solution of a related subnormal Toeplitz completion problem.Many problems in physics, mathematics, and engineering can be best described by representing complex physical entities as large arrays of numbers and mathematical symbols, called matrices. Matrices help us visualize how linear transformations act on vector spaces; determining their structure reveals important properties of the transformations. Hilbert space operators are infinite-dimensional generalizations of matrices. The generalization of a vector is often a function, and as a result, operators are frequently modeled as multiplications on spaces of functions. Part of this project involves finding such models for operators. Once the models are obtained, many basic structural questions about the operators become natural. Beginning in the 1950s, the study of subnormal operators has been highly successful, and its theory has made key contributions to areas such as functional analysis, quantum mechanics, and engineering. Similarly, the classes of scalar and block Toeplitz operators arise in a variety of areas of mathematics and physics. A separate part of the research deals with inverse problems that occur naturally in statistics, spectral analysis, geophysics, image recognition, global positioning tools, signal detection theory, and economics. The principal investigator's work on truncated moment problems has been applied in optimization theory, real algebraic geometry, numerical analysis, semidefinite programming, and sensor network localization. The project aims to resolve some outstanding problems in multivariable operator theory, at the same time creating recruitment and retention opportunities for women and minorities to pursue careers in mathematics and other STEM fields. Several questions in this project are written to generate research problems accessible to undergraduate and graduate students.