Multivariable Operator Theory

多变量算子理论

• 批准号: 9800931
• 负责人: Raul Curto
• 金额: $ 16.02万
• 依托单位: University of Iowa
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-06-01 至 2002-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9800931&HistoricalAwards=false
• 关键词: Multivariable; Operator; Theory

To: Dr. Joe Jenkins From: Raul Curto Date: January 7, 1998 Re: Abstract of Project, Multivariable Operator Theory This research project deals with four areas of multivariable operator theory: (i) existence, uniqueness, and localization of the support of representing measures for truncated moment problems; (ii) structure and spectral theory for polynomially hyponormal operators; (iii) standard operator models over Reinhardt domains; and (iv) a multivariable analog of Apostol's Lemma on hyperinvariant subspaces for contractions. Concerning the first area, special emphasis will be given to the study of moment problems associated with singular matrices, building on recent work (joint with L.A. Fialkow) on flat extensions of positive moment matrices, which has led to a general framework for the study of truncated complex moment problems. With the aid of a new moment matrix associated to a semi-algebraic set, progress is expected in the localization-of-support problem. A characterization of quadrature domains in terms of their moment matrices, and further applications to quadrature and cubature formulas will also be sought. As part of (ii), a characterization of polynomially hyponormal weighted shifts, a structure theorem for quadratically hyponormal shifts, and the detection of non-subnormal polynomially hyponormal operators through the Pincus principal function and through Putinar's 2-subscalar model, will be sought. The research aim in the third area (which deals with a Sz. Nagy- Foias dilation theory in several variables and builds on recent work of A. Athavale, V. Muller, F.-H. Vasilescu and others), is to extend the existing theory to functional Hilbert spaces over Reinhardt domains. The suitability of multi-shifts as standard models, the validity of von Neumann's inequality for special n-tuples, and the structure of C*-algebras generated by spherical isometries, will be considered. The fourth and final area deals with the invariant subspace structure of commuting contractions on Hilbert space. Using recent results of J. Eschmeier, M. Kosiek, A. Octavio, M. Ptak, and others, two main goals will be pursued: (a) an extension to the Taylor spectrum of results on spectral dominance currently available only for the Harte spectrum, and (b) an analog of Apostol's Theorem in several variables. Multivariable operator theory is a rapidly evolving area of mathematics, with deep and significant connections with areas of differential geometry, topology, complex analysis, and algebraic geometry, and with exciting applications to engineering, quantum and relativistic mechanics, and computational mathematics. The theory of truncated moment problems provide easily accessible formulas for the evaluation of areas and volumes of complex regions, of moments of inertia and centers of gravity. Dilation theory and invariant subspace theory are essential tools to describe the algebraic properties of elaborate physical or engineering systems, and the study of transformations on function spaces has often led to the solution of problems in control theory, intimately tied to systems theory and electrical engineering. Our research project is aimed at resolving some outstanding problems in multivariable operator theory, while creating recruitment and retention opportunities for women and minorities to pursue careers in mathematics, by engaging their participation in projects related to the interaction of mathematics with other sciences.

致：Joe Jenkins 博士 发件人：Raul Curto 日期：1998 年 1 月 7 日 回复：项目摘要，多变量算子理论 该研究项目涉及多变量算子理论的四个领域：(i) 存在性、唯一性以及支持的局部性代表截断矩问题的度量； (ii) 多项式次正规算子的结构和谱理论； (iii) Reinhardt 域上的标准算子模型； (iv) Apostol 引理关于收缩超不变子空间的多变量模拟。 关于第一个领域，将特别强调与奇异矩阵相关的矩问题的研究，以最近（与 L.A. Fialkow 联合）关于正矩矩阵的平坦扩展的工作为基础，该工作形成了研究截断复杂矩问题。 借助与半代数集相关的新矩矩阵，支撑定位问题有望取得进展。 还将寻求用矩矩阵来描述正交域的特征，以及对正交和立方公式的进一步应用。 作为 (ii) 的一部分，将寻求多项式次正规加权移位的表征、二次次次正规移位的结构定理以及通过 Pincus 主函数​​和 Putinar 的 2-子标量模型检测非次正规多项式次正规算子。 第三个领域的研究目标（涉及多个变量的 Sz. Nagy- Foias 膨胀理论，并建立在 A. Athavale、V. Muller、F.-H. Vasilescu 等人最近的工作基础上），是扩展Reinhardt 域上的函数希尔伯特空间的现有理论。 将考虑多位移作为标准模型的适用性、冯·诺依曼不等式对于特殊 n 元组的有效性以及由球面等距生成的 C* 代数的结构。 第四个也是最后一个区域处理希尔伯特空间上通勤收缩的不变子空间结构。 利用 J. Eschmeier、M. Kosiek、A. Octavio、M. Ptak 等人的最新结果，将实现两个主要目标：(a) 对目前仅适用于 Harte 的谱优势结果的泰勒谱进行扩展谱，以及（b）阿波斯托尔定理在几个变量中的模拟。 多变量算子理论是一个快速发展的数学领域，与微分几何、拓扑、复分析和代数几何领域有着深刻而重要的联系，并且在工程、量子和相对论力学以及计算数学中有着令人兴奋的应用。 截断力矩问题理论为评估复杂区域的面积和体积、惯性矩和重心提供了易于使用的公式。膨胀理论和不变子空间理论是描述复杂物理或工程系统的代数性质的重要工具，对功能空间变换的研究常常导致控制理论问题的解决，与系统理论和电气工程密切相关。 我们的研究项目旨在解决多变量算子理论中的一些突出问题，同时通过让女性和少数族裔参与与数学与其他科学相互作用相关的项目，为女性和少数族裔从事数学职业创造招聘和保留机会。