Extending Hilbert Space Operators

扩展希尔伯特空间算子

• 批准号: 0801259
• 负责人: Jim Agler
• 金额: $ 29.15万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-07-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0801259&HistoricalAwards=false
• 关键词: Extending; Hilbert; Space; Operators

Our research proposal concerns the interplay between Operator Theory and Function Theory in one and several variables. A particular group of problems that we propose to attack involves the generalizations to several complex variables of some of the classical moment and interpolation problems on the unit disc such as the interpolation theorem of Nevanlinna and Pick, and the moment theorems of Caratheodory and Herglotz. Other groups involve the generalization of geometric function theory to analytic varieties and the understanding of extremal mappings into the symmetrized bidisc. Research intrinsic to operator theory that we will undertake includes issues involving model theory in one variable on nonsimply connected domains in the plane and in several variables on domains other than the bidisc. Operator Theoretic Function Theory, the particular type of mathematics that we are proposing to investigate, has direct and concrete benefits for a number of areas of human endeavor. For example, the model theory aspects of our proposal all involve the generalization of the Commutant Lifting Structure which leads to an efficient algorithm for the discovery of oil from acoustical data taken on the surface of the earth. Other aspects would add to the theory of Linear Matrix Inequalities. The mathematical theory of LMI?s, which currently is of great importance in several areas of engineering, is an extension of linear programming, which has made large scale resource allocation and economic prediction possible. Finally, the particular brand of function theory we propose to study, forms the mathematical core of H-infinity control theory, which has been used to design control systems for fusion reactions inside Tokamaks and feedback stabilization systems for the space shuttle.

我们的研究计划涉及算子理论和函数理论在一个和几个变量中的相互作用。我们提出要解决的一组特殊问题涉及到单位圆盘上的一些经典矩和插值问题的几个复变量的推广，如Nevanlinna和Pick的插值定理，以及Caratheodory和Herglotz的矩定理。其他小组涉及几何函数理论的推广到解析变分和极值映射到对称二似体的理解。我们将进行算子理论固有的研究，包括平面上非单连通域上的一个变量模型理论和双曲面以外的域上的几个变量模型理论。算符理论函数理论是我们要研究的特殊数学类型，它有直接和具体的好处。它适用于人类努力的许多领域。例如，我们提案的模型理论方面都涉及到换向提升结构的泛化，从而导致e&amp；#64259；从地球表面的声学数据中发现石油的客户算法。其他方面将增加线性矩阵不等式的理论。LMI的数学理论？S是线性规划的一种扩展，它使大规模的资源分配和经济预测成为可能，目前在工程的多个领域都具有重要意义。最后，我们要研究的函数理论的特定品牌，形成了h -in的数学核心。统一控制理论，它被用来设计托卡马克内部聚变反应的控制系统和航天飞机的反馈稳定系统。