Multivariable Operator Theory

多变量算子理论

• 批准号: 0801168
• 负责人: Raul Curto
• 金额: $ 15万
• 依托单位: University of Iowa
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-06-01 至 2013-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0801168&HistoricalAwards=false
• 关键词: Multivariable; Operator; Theory

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.

摘要 Curto 该研究涉及多变量算子理论，重点关注三个领域：(i) 截断矩问题 (TMP) 表示测度支持的代数条件、唯一性和局部性；(ii) 次正规性检测中的多变量技术； ，特别是单位圆上的 Toeplitz 算子，包括通勤次正规数提升问题 (LPCS) 的方法；以及 (iii) Reinhardt 域上的算子理论，特别关注多变量加权位移的谱和结构特性，我们计划扩展最近关于正矩矩阵平面扩展的工作（与 L. Fialkow 和 H.M. Möller 合作），这导致了一个通用的问题。我们计划将这些方法应用于极端情况之外，以获得溶解度的代数和几何不变量，以进一步开发适当的类似物。 Riesz-Haviland 定理，并研究 TMP 与指定半代数集上非负多项式的有界表示之间的对偶性。第二个领域涉及 LPCS 的多变量方法以及基于 C. Cowen 算子的次正规性。对于次正规 Toeplitz 算子的情况，我们的方法是首先表征 2-次正规性，然后是 k-次正规性，最后是次正规性。还想进一步发展最近与 J. Yoon 和 S.H. Lee 的联合工作中的想法，以寻找两个可交换次正规算子承认联合法线扩展的必要和充分条件，包括与 Agler 的抽象模型理论的一些有用的联系。处理 Reinhardt 域上函数希尔伯特空间上的乘法算子的结构和谱特性，我们计划利用最近的结果（与J. Yoon）强调了当伯杰测度不存在时出现的一些病态，并且使用与 P. Muhly 联合工作中引入的群形技术是矩阵的无限泛化。函数，因此希尔伯特空间运算符经常被建模为函数空间上的乘法运算符，该项目的一部分涉及找到运算符或运算符元组的此类模型，一旦获得此类模型，就会出现有关结构的许多基本问题。这些算子的一部分变得更加自然。研究的一个单独部分涉及反问题，特别是与质量分布的幂矩相关的问题，并且自然出现在统计学、光谱分析、地球物理学、图像识别和经济学中。我们的研究旨在解决多变量算子理论中的一些突出问题，同时通过参与与数学与其他科学相互作用相关的项目，为女性和少数族裔创造招聘和保留机会以从事数学职业。片刻S. McCullough 使用该问题获得了 Fejér-Riesz 分解理论中的结构定理；J. Lasserre 在平面的半代数子集的研究中使用了该问题；J. Lasserre 和 M. Laurent 将多项式优化转化为我们预计，本提案中的几个开放问题将继续出现，以产生可供本科生和研究生使用的研究项目，特别是与算子理论相关的项目。立方体、低次矩问题、它们与代数几何的联系以及多变量加权平移。