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（称为矩阵）来描述。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（考虑无限大小）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。该项目的一个主要目标是为运营商找到这样的模型。一旦做到这一点，许多基本的结构性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。其目的是解决所谓的多变量算子理论中的几个突出问题。该项目还涉及与学生合作，特别是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。 这个项目的主要思想/推力将是利用最近建立的解析几何和分析之间的联系，研究多变量算子理论的问题。注意力将集中在两个主要领域：（一）截断矩问题（TMP）;（二）（联合）亚正规性和次正规性的交换家庭的运营商在希尔伯特空间。在第一个领域中，代数条件将被确定的存在性，唯一性和本地化的支持代表措施TMP。在矩量矩阵与列关系的情况下，新的溶解度准则的发展将被绑定到与有限代数簇相关的不可约代数曲线和立方列关系。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移位的谱和结构性质。计划都是研究一个新的桥梁连接2变量加权移位理论的加权移位有向树，和表征的时刻无限可分加权移位，使用理论的完全交替序列和完全单调函数，伯恩斯坦函数，和拉普拉斯和傅立叶变换。计划中的方法将包括多变量技术在块Toeplitz operators.This奖项的研究反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。