Multivariable Operator Theory

摘要本研究涉及多变量算子理论，主要关注三个方面：(i)截断矩问题（TMP）表示测度的存在性、唯一性和局部性的代数条件；（ii）次正态检测中的多变量技术，特别是对于单位圆上的Toeplitz算子，包括交换次正态的提升问题（LPCS）的一种方法；（iii） Reinhardt域上的算子理论，特别关注多变量加权移位的谱和结构性质。关于第一个领域，我们计划扩展最近关于正矩矩阵的平面扩展的工作（与L. Fialkow和H.M. Möller联合），这导致了TMP研究的一般框架。我们计划将这些方法应用于极值情况之外，以获得溶解度的代数和几何不变量，进一步发展Riesz-Haviland定理的适当模拟，并研究TMP与规定半代数集上非负多项式的度有界表示之间的对偶性。第二个领域涉及LPCS的多变量方法和Toeplitz算子的次正态性。基于C. Cowen对次非正常Toeplitz算子的研究，我们的方法是首先描述2-次非正常，然后是k-次非正常，最后是次非正常。我们还将进一步发展最近与J. Yoon和S.H. Lee共同研究的关于寻找两个可交换次正规算子允许联合正规扩展的充分必要条件的思想，包括与Agler抽象模型理论的一些有用联系。第三部分研究了Reinhardt域上泛函Hilbert空间上乘法算子的结构和谱性质。我们计划利用最近的结果（与J. Yoon联合），利用与P. Muhly联合工作中引入的群样技术，将亚正态多变量加权移位的谱图研究扩展到次正态的谱图。这些结果突出了Berger测量缺失时出现的一些病理。希尔伯特空间算子是矩阵的无限推广。向量的无限泛化通常是一个函数，因此希尔伯特空间算子通常被建模为函数空间上的乘法算子。这个项目的一部分涉及为操作符或操作符元组找到这样的模型。一旦获得了这样的模型，关于这些算子结构的许多基本问题就变得更加自然了。研究的另一部分涉及逆问题，特别是矩问题，这与质量分布的幂矩有关，并且在统计学，光谱分析，地球物理学，图像识别和经济学中自然出现。我们的研究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族裔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。S. McCullough利用截断矩问题的结果，得到了fej<s:1> - riesz分解理论中的一个结构定理；J. Lasserre在平面半代数子集的研究；由J. Lasserre和M. Laurent将多项式优化转化为半定规划的实例。我们预计，这种与算子理论以外领域的联系将继续出现。本提案中的几个开放问题旨在为本科生和研究生提供可访问的研究项目，特别是与培养，低次矩问题，它们与代数几何的联系以及多变量加权移位相关的问题。

AbstractCurtoThe research deals with multivariable operator theory, focusing attention on three areas: (i) algebraic conditions for existence, uniqueness, and localization of the support of representing measures for truncated moment problems (TMP); (ii) multivariable techniques in the detection of subnormality, esp. for Toeplitz operators on the unit circle, including an approach to the Lifting Problem for Commuting Subnormals (LPCS); and (iii) operator theory over Reinhardt domains, with special attention given to the spectral and structural properties of multivariable weighted shifts. Concerning the first area, we plan to extend recent work on flat extensions of positive moment matrices (joint with L. Fialkow and H.M. Möller), which has led to a general framework for the study of TMP. We plan to apply these methods beyond the extremal case, to obtain algebraic and geometric invariants for solubility, to further develop an appropriate analogue of the Riesz-Haviland Theorem, and to investigate the duality between TMP and degree-bounded representations of polynomials nonnegative on a prescribed semialgebraic set. The second area deals with a multivariable approach to LPCS and with subnormality for Toeplitz operators. Building on work of C. Cowen for the case of hyponormal Toeplitz operators, our approach is to first characterize 2-hyponormality, then k-hyponormality, and eventually subnormality. We would also like to develop further the ideas in recent joint work with J. Yoon and S.H. Lee to search for necessary and sufficient conditions for two commuting subnormal operators to admit a joint normal extension, including some useful connections with Agler's abstract model theory. The third area deals with structural and spectral properties of multiplication operators on functional Hilbert spaces over Reinhardt domains. We plan to extend the study of the spectral picture of subnormal multivariable weighted shifts to hyponormal ones, exploiting recent results (joint with J. Yoon) which highlight some of the pathology that arises when a Berger measure is absent, and using the groupoid techniques introduced in joint work with P. Muhly.Hilbert space operators are infinite generalizations of matrices. The infinite generalization of a vector is frequently a function and for this reason Hilbert space operators are frequently modeled as the operator of multiplication on a space of functions. Part of this project involves finding such models for operators or tuples of operators. Once such models are obtained many basic questions about the structure of these operators become more natural. A separate part of the research deals with inverse problems, esp. moment problems, which are related to power moments of mass distributions, and arise naturally in statistics, spectral analysis, geophysics, image recognition, and economics. Our research 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. The results on truncated moment problems have been used by S. McCullough to obtain a structure theorem in Fejér-Riesz factorization theory; by J. Lasserre in the study of semi-algebraic subset of the plane; and by J. Lasserre and M. Laurent to convert polynomial optimization into an instance of semidefinite programming. We anticipate that such connections with areas outside of operator theory will continue to arise. Several open problems in this proposal are written to generate research projects accessible to undergraduate and graduate students, especially those related to cubatures, low-degree moment problems, their connections with algebraic geometry, and multivariable weighted shifts.