Multivariable Operator Theory

本研究计划涉及多变量算子理论的五个领域：（i）截断矩问题表示测度支集的存在性、唯一性和局部化的代数条件;（ii）2-亚正规算子的模型理论;（iii）多项式亚正规算子的结构和谱理论;（iv）Reinhardt域和矩阵Reinhardt域上的算子模型;（v）收缩超不变子空间上Apostol引理的多变量模拟。关于第一个领域，Curto计划扩展他最近的工作（与L. Fialkow）关于正矩矩阵的平坦扩张，为截断复矩问题的研究提供了一个一般框架。他计划应用这些方法来研究奇异矩矩阵的情况下，并继续分析问题的本地化的支持代表措施。Curto期望解决截断矩序列的溶解度指数的猜想，并根据矩矩阵来表征正交域。作为（ii）的一部分，将寻求2-亚正规算子的模型理论，沿着现有的亚正规算子理论的路线，并使用单边加权移位的最新结果作为试验场。与W. Y. Lee，Curto引进了弱次正规算子类，并得到了它相对于次正规性和2-亚正规性的位置的初步结果，包括证明了具有闭值域自换位子的压缩2-亚正规算子是2-亚正规压缩算子族的极值.第三个领域是密切相关的前两个，在这两个起源于Curto的工作二次和联合亚正规性，这最终导致解决的次正常完成问题的加权移位和存在的非次正常多项式亚正规运营商。Curto将寻求多项式次正规加权移位的表征，二次正规移位的结构定理，以及通过Pincus主函数检测非次正规多项式次正规算子。第四个领域涉及一个SZ。将已有的结果推广到Reinhardt域上的泛函Hilbert空间，得到了多变量的Nagy-Foias膨胀理论. 将考虑多重移位作为标准模型的适用性和冯·诺依曼不等式对特殊n元组的有效性。Curto计划扩展与莱因哈特措施功能希尔伯特空间矩阵莱因哈特域的乘法算子的谱图的描述。最后，第五部分讨论了Hilbert空间上交换压缩的不变子空间结构。将追求两个主要目标：（a）扩展到泰勒频谱的频谱优势的结果，目前只可用于哈特频谱，和（B）模拟阿波斯托尔定理在几个变量。多变量算子理论是一个快速发展的数学领域，与微分几何，拓扑学，复分析和代数几何领域有着深刻而重要的联系，并在工程，量子和相对论力学以及计算数学中有着令人兴奋的应用。截断矩问题的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方便的公式。膨胀理论和不变子空间理论是描述复杂的物理或工程系统的代数性质的重要工具，并且函数空间上的变换的研究经常导致控制理论中的问题的解决，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，同时为妇女和少数民族创造招聘和保留机会，通过参与与数学与其他科学相互作用有关的项目，从事数学职业。

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.