Extending Hilbert Space Operators

扩展希尔伯特空间算子

• 批准号: 0400826
• 负责人: Jim Agler
• 金额: $ 26.26万
• 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: 2004
• 资助国家: 美国
• 起止时间: 2004-07-01 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0400826&HistoricalAwards=false
• 关键词: Extending; Hilbert; Space; Operators

ABSTRACTAglerAfter the spectral theorem it is difficult to think of a theorem that has had a more profound effect on the development of operator theory and its myriads of applications to mathematics and science than the Sz.-Nagy Dilation Theorem. The idea of representing a general operator in a specialized class of operators as a part of a nice operator in the class has had many successes and we seek to develop this point of view with a primary focus on applications to problems in the theory of functions in one and several complex 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. Another focus of our project will be the use of operator-theoretic methods to study function theory on analytic varieties. 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. This will include the continuation of our work on the symmetrized bidisc with a particular focus on applications to complex geometry and the mu-synthesis problem.Operator Theory, the particular type of mathematics that we are proposing to develop, 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 enrich the theory of Linear Matrix Inequalities. LMI 's, which currently are all the rage in several areas of engineering, are an extension of linear programming, a mathematics which has made possible not only the optimization of large scale resource allocation but the accurate prediction of economic markets as well. Finally, the particular branch of function theory we propose to study, forms the mathematical core of the 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.

在谱定理之后，很难想到还有什么定理比Sz.-Nagy膨胀定理对算子论的发展及其在数学和科学中的无数应用产生了更深远的影响。将一个特殊的算子类中的一个一般算子表示为该类中一个好算子的一部分的想法已经取得了许多成功，我们试图发展这一观点，主要集中在应用于一元多复变量函数论中的问题。我们提出要解决的一组特殊问题涉及到单位圆盘上一些经典的矩和内插问题的几个复变量的推广，如Nevanlinna和Pick的插值定理以及Caratheodory和Hergltz的矩定理。我们项目的另一个重点将是使用算子论的方法来研究解析簇上的函数理论。我们将承担的算子理论的内在研究包括涉及平面上非单连通区域上的一个变量的模型理论以及除双圆盘以外的区域上的多个变量的模型理论。这将包括继续我们在对称双盘上的工作，特别关注于复杂几何和单位合成问题的应用。运算符理论，我们建议发展的特定类型的数学，对人类努力的许多领域具有直接和具体的好处。例如，我们建议的模型理论方面都涉及到对交换提升结构的推广，这导致了从地球表面的声学数据中发现石油的有效算法。其他方面将丰富线性矩阵不等式的理论。线性规划S是线性规划的推广，它不仅使大规模资源配置的优化成为可能，也使经济市场的准确预测成为可能。最后，我们提出研究的函数论的一个特殊分支构成了H无穷控制理论的数学核心，该理论已被用于设计托卡马克内聚变反应的控制系统和航天飞机的反馈稳定系统。