Multivariable Operator Theory

本研究项目涉及多变量算子理论的五个领域：(i)截断矩问题表示测度的存在性、唯一性和局域性支持的代数条件；（ii） 2-次正规算子的模型理论；（iii）多项式次正规算子的结构和谱理论；（iv） Reinhardt和矩阵Reinhardt域上的算子模型；(v)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，Curto计划扩展他最近的工作（与L. Fialkow联合）关于正矩矩阵的平面扩展，这导致了截断复矩问题研究的一般框架。他计划将这些方法应用到奇异矩矩阵的研究中，并继续分析表示测度支持的局部化问题。Curto期望解决截断矩序列的溶解度指数的猜想，并根据它们的矩矩阵来表征正交域。作为（ii）的一部分，将沿着现有的次正规算子理论的路线寻求2-次正规算子的模型理论，并使用单边加权位移的最新结果作为试验场。在与W.Y. Lee的联合工作中，Curto引入了弱次正规算子类，并得到了它相对于次正规和2-次正规的位置的初步结果，包括证明了具有闭范围自换子的压缩2-次正规算子对于2-次正规缩族是极值的。第三个领域与前两个领域密切相关，因为它们都源于Curto关于二次和联合次非正常的工作，最终导致了加权移位的次正规补全问题的解决以及非次正规多项式次非正常算子的存在。Curto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主函数检测非次正规多项式次正规算子。第四个领域涉及Sz。通过将已有结果推广到Reinhardt域上的泛函Hilbert空间，在若干变量下的Nagy-Foias膨胀理论。本文将考虑多位移作为标准模型的适用性以及von Neumann不等式对特殊n元组的有效性。Curto计划将与Reinhardt测度相关的乘法算子谱图的描述扩展到矩阵Reinhardt域上的泛函Hilbert空间。最后，第五部分讨论Hilbert空间上交换收缩的不变子空间结构。本研究的两个主要目标是：(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)在几个变量中类比阿波斯托尔定理。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。

This research project deals with five areas of multivariable operator theory: (i) algebraic conditions for existence, uniqueness, and localization of the support of representing measures for truncated moment problems; (ii) model theory for 2-hyponormal operators; (iii) structure and spectral theory for polynomially hyponormal operators; (iv) operator models over Reinhardt and matrix Reinhardt domains; and (v) a multivariable analog of Apostol's Lemma on hyperinvariant subspaces for contractions. Concerning the first area, Curto plans to extend his recent work (joint with L. Fialkow) on flat extensions of positive moment matrices, which has led to a general framework for the study of truncated complex moment problems. He plans to apply these methods to study the case of singular moment matrices, and to continue to analyze the question of localization of the support of representing measures. Curto expects to settle a conjecture on the solubility index of a truncated moment sequence, and to characterize quadrature domains in terms of their moment matrices. As part of (ii), a model theory for 2-hyponormal operators will be sought, along the lines of the existing theory for subnormal operators, and using as test ground recent results on unilateral weighted shifts. In joint work with W.Y. Lee, Curto has introduced the class of weakly subnormal operators and obtained preliminary results on its position relative to subnormality and 2-hyponormality, including a proof that contractive 2-hyponormal operators with closed range self-commutator are extremal for the family of 2-hyponormal contractions. The third area is closely related to the first two, in that both originate in Curto's work on quadratic and joint hyponormality, which eventually led to the solution of the subnormal completion problem for weighted shifts and to the existence of non-subnormal polynomially hyponormal operators. Curto will seek 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. The fourth area deals with a Sz. Nagy-Foias dilation theory in several variables, by extending existing results to functional Hilbert spaces over Reinhardt domains. The suitability of multi-shifts as standard models and the validity of von Neumann's inequality for special n-tuples will be considered. Curto plans to extend the description of the spectral picture of multiplication operators associated with Reinhardt measures to functional Hilbert spaces over matrix Reinhardt domains. Finally, the fifth area deals with the invariant subspace structure of commuting contractions on Hilbert space. 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 in the description of 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.