Multivariable Operator Theory

本文研究了多变量算子理论，主要集中在三个方面：（1）截断矩问题（TMP）表示测度支集的存在性、唯一性和局部性的代数条件;（2）多变量次正规性检测技术，特别是单位圆上Toeplitz算子的次正规性检测技术，包括交换次正规性提升问题（LPCS）的方法; 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，我们计划扩展最近的工作，平坦扩张的正矩矩阵（联合L。Fialkow和H.M. Möller），这导致了TMP研究的一般框架。 我们计划应用这些方法超出极端情况下，获得代数和几何不变的溶解度，进一步开发一个适当的模拟的Riesz-Haviland定理，并调查TMP和度之间的对偶有界表示多项式非负的一个规定的半代数集。 第二个领域涉及一个多变量的方法来LPCS和次正态的Toeplitz运营商。 在C. Cowen对于亚正规Toeplitz算子，我们的方法是首先刻画2-亚正规性，然后刻画k-亚正规性，最后刻画次正规性。 我们还希望在最近与J. Yoon和S.H.李寻找必要和充分条件，两个交换subnormal运营商承认一个联合正常的扩张，包括一些有用的连接与Agler的抽象模型理论。 第三部分研究Reinhardt域上泛函Hilbert空间上乘法算子的结构和谱性质。 我们计划扩大研究的频谱图片的次正常的多变量加权移位亚正常的，利用最近的结果（联合J.尹），突出了一些病理时出现的伯杰措施是缺席，并使用广群技术介绍了联合工作与P. Muhly。希尔伯特空间算子是无限推广的矩阵。 向量的无限推广通常是一个函数，因此希尔伯特空间算子经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到为运算符或运算符元组找到这样的模型。 一旦获得了这样的模型，关于这些算子的结构的许多基本问题就变得更加自然了。 研究的一个单独部分涉及逆问题，特别是矩问题，它与质量分布的幂矩有关，并且在统计学，光谱分析，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悬而未决的问题，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，为她们创造从事数学职业的招聘和保留机会。 截断矩问题的结果已被S. McCullough在Fejér-Riesz分解理论中得到的一个结构定理; J.拉瑟尔在研究平面的半代数子集时得到的;以及J.拉瑟尔和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的领域的联系将继续出现。 本建议中的几个开放问题是为了生成本科生和研究生可以访问的研究项目而编写的，特别是那些与cubatures，低阶矩问题，它们与代数几何的联系以及多变量加权移位有关的问题。

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.