Multivariable Operator Theory

多变量算子理论

• 批准号: 1302666
• 负责人: Raul Curto
• 金额: $ 19.36万
• 依托单位: University of Iowa
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-08-01 至 2017-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1302666&HistoricalAwards=false
• 关键词: Multivariable; Operator; Theory

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.

本研究涉及多变量算子理论，重点关注三个领域：（一）代数条件的存在性，唯一性和本地化的支持表示措施截断矩问题（TMP）;（二）运营商理论在莱因哈特域，特别强调频谱和结构特性的多变量加权移位;和（三）多变量技术在块Toeplitz运营商的研究。 关于第一个领域，计划是扩展最近的工作平坦扩展的正矩矩阵和极值矩问题（联合L。Fialkow，H.M. Möller和S. Yoo），这导致了TMP研究的一般框架。 主要研究者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簇相关的立方列关系的情况，以及具有递归确定矩矩阵的TMP。 第二个领域涉及Reinhardt域上泛函Hilbert空间上的乘法算子。 这一部分的项目将扩大研究的频谱图片的亚正规2变量加权移位亚正规的，通过应用以前的结果（联合S.H. Lee和J. Yoon），并采用主要研究者与P. Muhly和K.燕. 第三个领域涉及一个多变量的方法，次正规的希尔伯特空间运营商，特别强调标量和块Toeplitz运营商。 方法是先刻画2-亚正规性，然后刻画k-亚正规性，最后刻画次正规性。 首席研究员将进一步发展他以前与I.B.合作的想法。荣格，S.H. Lee，W.Y. Lee，S.S. Park和M. Putinar. 作为一个试验场，他将寻找一个模型理论的2-亚规范运营商，一个主题，导致有用的连接与J.阿格勒的抽象模型理论。 利用函数理论和多变量算子理论技术，主要研究者最近证明了块Toeplitz算子的Abrahamse定理的一个版本（与I.S. Hwang和W.Y. Lee）。 物理学、数学和工程学中的许多问题都可以通过将复杂的物理实体表示为大量的数字和数学符号的数组（称为矩阵）来最好地描述。 矩阵帮助我们形象化线性变换如何作用于向量空间;确定它们的结构揭示了变换的重要性质。 希尔伯特空间算子是矩阵的无限维推广。 向量的推广通常是一个函数，因此，运算符经常被建模为函数空间上的乘法。 该项目的一部分涉及为运营商找到这样的模型。 一旦得到了模型，许多关于算子的基本结构问题就变得很自然了。 从20世纪50年代开始，亚正规算子的研究取得了巨大的成功，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算子的类出现在数学和物理的各个领域。 研究的一个单独的部分涉及自然发生在统计，频谱分析，物理学，图像识别，全球定位工具，信号检测理论和经济学中的逆问题。 主要研究人员的工作截断矩问题已被应用于优化理论，真实的代数几何，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，同时为妇女和少数民族创造招聘和保留机会，以追求数学和其他STEM领域的职业生涯。 在这个项目中的几个问题被写来生成本科生和研究生可以访问的研究问题。