Multivariable Operator Theory

多变量算子理论

• 批准号: 2247167
• 负责人: Raul Curto
• 金额: $ 24.78万
• 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: 2023
• 资助国家: 美国
• 起止时间: 2023-08-01 至 2026-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2247167&HistoricalAwards=false
• 关键词: Multivariable; Operator; Theory

Many questions in physics, mathematics, and engineering can be 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 (think infinite-size) 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. A main goal of this project involves finding such models for operators. Once that is done, many basic structural questions 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. The aim is to resolve several outstanding questions in so-called multivariable operator theory. The project also involves working with students and creating recruitment and retention opportunities particularly for women and minorities to pursue careers in mathematics and other STEM fields. The main idea/thrust of this project will be to utilize recently established connections between analytic geometry and analysis to study questions in multivariable operator theory. Attention will be focused on two principal areas: (i) truncated moment problems (TMP); and (ii) (joint) hyponormality and subnormality for commuting families of operators on Hilbert space. In the first area, algebraic conditions will be determined for the existence, uniqueness, and localization of the support of representing measures for TMP. Development of new solubility criteria in the case of moment matrices with column relations is to be tied to irreducible algebraic curves and cubic column relations associated with finite algebraic varieties. The second direction concerns operator theory over Reinhardt domains, with special emphasis on spectral and structural properties of multivariable weighted shifts. Planned are both the study of a new bridge connecting 2-variable weighted shifts to the theory of weighted shifts on directed trees, and the characterization of moment infinitely divisible weighted shifts, using the theory of completely alternating sequences and completely monotone functions, Bernstein functions, and Laplace and Fourier transforms. The planned methodology will include multivariable techniques in the study of block Toeplitz operators.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

物理、数学和工程中的许多问题可以通过将复杂的物理实体表示为大量数字和数学符号（称为矩阵）来描述。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属性。希尔伯特空间运算符是矩阵的无限维（认为无限大小）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。该项目的主要目标是为运营商找到这样的模型。一旦完成，许多基本的结构问题就变得自然了。从20世纪50年代开始，次正规算子的研究取得了巨大成功，其理论对泛函分析、量子力学和工程学等领域做出了重要贡献。目的是解决所谓的多变量算子理论中的几个悬而未决的问题。该项目还涉及与学生合作，创造招聘和保留机会，特别是为女性和少数族裔从事数学和其他 STEM 领域的职业。 该项目的主要思想/主旨是利用解析几何和分析之间最近建立的联系来研究多变量算子理论中的问题。注意力将集中在两个主要领域：（i）截断矩问题（TMP）； (ii) 希尔伯特空间上算子通勤族的（联合）次正规性和次正规性。在第一个领域，将确定 TMP 表示测度支持的存在性、唯一性和局部性的代数条件。在具有列关系的矩矩阵的情况下，新的溶解度准则的开发将与与有限代数簇相关的不可约代数曲线和立方列关系联系在一起。第二个方向涉及莱因哈特域上的算子理论，特别强调多变量加权平移的谱和结构特性。计划研究连接 2 变量加权移位与有向树加权移位理论的新桥梁，以及使用完全交替序列和完全单调函数、伯恩斯坦函数的理论来表征矩无限可分加权移位。以及拉普拉斯和傅立叶变换。计划的方法将包括对 Toeplitz 区块运营商的研究中的多变量技术。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力优点和更广泛的影响审查标准进行评估，被认为值得支持。