Multivariable Operator Theory

致：Dr. Joe Jenkins来自：Raul Curto日期：1998年1月7日Re：项目摘要，多变量算子理论本研究项目涉及多变量算子理论的四个领域：(i)截断矩问题表示测度支持的存在性、唯一性和局域性；（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）缩缩超不变子空间上Apostol引理的一个多变量类比。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）关于正矩矩阵的平面扩展的工作为基础，这导致了截断复矩问题研究的一般框架。借助与半代数集相关联的新的矩矩阵，期望在支持定位问题上取得进展。在矩矩阵方面的正交域的特征，以及进一步的应用于正交和培养公式也将寻求。作为（ii）的一部分，将寻求多项式次正规加权位移的表征，二次次正规位移的结构定理，以及通过Pincus主函数和Putinar的2-次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的是研究一个Sz。Nagy- Foias在几个变量中的膨胀理论，并建立在A. Athavale， V. Muller, f . h .的最新工作基础上。Vasilescu等)，是将现有理论推广到Reinhardt域上的泛函Hilbert空间。将考虑多位移作为标准模型的适用性，von Neumann不等式对特殊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讨论Hilbert空间上交换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，我们将追求两个主要目标：(a)将目前仅适用于哈特谱的谱优势性结果扩展到泰勒谱，以及(b)在几个变量中类比Apostol定理。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、复杂分析和代数几何等领域有着深刻而重要的联系，并在工程、量子力学和相对论力学以及计算数学中有着令人兴奋的应用。截断矩问题的理论为复杂区域的面积和体积、转动惯量和重心的计算提供了易于理解的公式。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基本工具，对函数空间变换的研究经常导致控制理论中问题的解决，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民族参与与数学与其他科学相互作用相关的项目，为她们在数学领域的职业发展创造招聘和保留机会。

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.