Multivariable Operator Theory

致：Joe Jenkins博士发件人：Raul Curto日期：1998年1月7日回复：多变量算子理论项目摘要 本研究计画探讨多元算子理论的四个领域：（i）多变量算子理论的存在性、唯一性和局部化 表示截断矩问题的措施;（ii）结构和 多项式亚正规算子谱理论;（iii）标准算子 Reinhardt域上的模型;以及（iv）Apostol的多变量模拟 关于压缩超不变子空间的引理。 就第一 区域，特别强调将给予时刻问题的研究 与奇异矩阵相关，建立在最近的工作（与L.A. Fialkow）关于正矩矩阵的平坦扩张，这导致了一个 截断复矩问题研究的一般框架。 借助于与半代数集相关联的新的矩矩阵， 预计在支助当地化问题上会取得进展。 一 用矩量矩阵表征正交域， 并进一步应用到求积和体积公式也将 寻找。 作为（ii）的一部分，多项式亚正规的特征 加权移位，二次亚正规移位的结构定理， 非次正规多项式次正规算子的检测 Pincus主函数和通过Putinar的2-subscalar模型，将是 寻找。 第三个领域的研究目标是：纳吉- Foias膨胀理论在几个变量和建立在最近的工作A。 Athavale，V. Muller，F. H. Vasilescu和其他人），是为了扩大现有的 Reinhardt域上的泛函Hilbert空间理论。 的适用性 作为标准模型，冯诺依曼不等式的有效性， 特殊的n-元组，以及球面生成的C*-代数的结构 将被考虑。 第四个也是最后一个领域涉及 Hilbert空间上交换压缩不变子空间结构 利用J. Jummeier，M. Kosiek，A. Octavio，M. Ptak，以及 在其他方面，将追求两个主要目标：（a）延长泰勒 频谱优势结果的频谱，目前仅适用于 Harte谱，和（B）在多个变量中Apostol定理的模拟。 多变量算子理论是一个快速发展的领域， 数学，与微分学领域有着深刻而重要的联系 几何学、拓扑学、复分析和代数几何，以及 令人兴奋的应用工程，量子和相对论力学，和 计算数学 截断矩问题的理论 提供易于使用的公式，用于评估 复杂的区域，惯性矩和重心。膨胀理论 和不变子空间理论是描述代数 复杂的物理或工程系统的性质，以及对 函数空间上的变换经常导致问题的解决 在控制理论中，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一些突出的问题， 多变量算子理论，同时创造招聘和保留 为妇女和少数民族提供从事数学职业的机会， 让他们参与与以下方面的互动有关的项目： 数学与其他科学

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.