Multivariable Operator Theory

致：Joe Jenkins博士来自：Raul Curto Date：1998年1月7日Re：项目摘要，多变量算子理论这项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的存在、唯一性和局部化；(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；以及(Iv)关于压缩超不变子空间的Apostol引理的多变量模拟。关于第一个领域，将特别强调与奇异矩阵有关的矩问题的研究，这是在最近(与L.A.Fialkow联合)关于正矩矩阵的平坦扩张的工作的基础上进行的，这导致了研究截断复矩问题的一般框架。借助于与半代数集相关的一个新的矩矩阵，支撑点的局部化问题有望取得进展。还将根据它们的矩矩阵来刻画求积域，并进一步应用于求积公式和体积公式。作为(Ii)的一部分，将寻求多项式次正规加权移位的刻画，二次正规移位的结构定理，以及通过Pincus主函数和Putina的2-次标量模型来检测非次正规多项式次正规算子。本研究的目的是在第三个领域(涉及一个SZ。Nagy-FOIA的多元膨胀理论，并建立在A.Athavale，V.Muller，F.-H.Vasilescu等人最近工作的基础上)，是将现有的理论推广到Reinhardt域上的泛函Hilbert空间。将考虑多重移位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。第四部分，也是最后一部分，讨论了Hilbert空间上交换压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，将追求两个主要目标：(A)推广目前仅适用于Harte谱的关于谱优势的Taylor谱结果，以及(B)在多个变量中类似于Apostol定理。多变量算符理论是一个快速发展的数学领域，与微分几何、拓扑学、复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。截断力矩理论为复杂区域的面积和体积、惯性矩和重心的计算提供了简单易用的公式。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基本工具，而对函数空间上的变换的研究往往导致控制理论中问题的解决，这与系统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符理论中的一些突出问题，同时通过让妇女和少数族裔参与与数学与其他科学相互作用的项目，为妇女和少数民族创造招聘和留住数学职业的机会。

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.