Extending Hilbert Space Operators

扩展希尔伯特空间算子

• 批准号: 0100607
• 负责人: Jim Agler
• 金额: $ 12.86万
• 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: 2001
• 资助国家: 美国
• 起止时间: 2001-07-01 至 2004-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0100607&HistoricalAwards=false
• 关键词: Extending; Hilbert; Space; Operators

ABSTRACT After 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 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 group involves deriving analogs of the theorem of Adamyan, Arov, and Krein on spaces more general than the classical Hardy space. 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 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 effcient 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. 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 recently developed 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.

在光谱定理之后的摘要很难想到一种对操作者理论的发展及其在数学和科学上的应用比SZ.-Nagy扩张定理更深远的定理。代表普通运营商在专业类中作为班级良好运营商的一部分的想法取得了许多成功，我们试图发展这一观点，主要关注一个和几个复杂变量的功能理论中的问题。我们提出攻击的一组特定问题涉及对单位光盘上一些经典时刻和插值问题的几个复杂变量的概括，例如Nevanlinna的插值定理和Pick and Pick以及Caratheodory和Herglotz的定理。另一组涉及得出比经典耐寒空间更通用的空间上Adamyan，Aroov和Kerin定理的类似物。我们将提出的固有的对操作者理论的固有理论包括在平面中非连接域的一个变量中涉及模型理论的问题，而在Bidisc以外的域上的几个变量中。操作者理论是我们建议研究的特定类型数学类型，它对人类努力的许多领域具有直接和具体的好处。例如，我们提案的模型理论方面均涉及换向提升结构的概括，这导致了一种从地球表面获取的声学数据发现油的效率算法。其他方面将增加线性矩阵不平等的理论。 LMI的目前在工程的几个领域中都流行，是线性编程的延伸，这是一种数学，这不仅使大规模资源分配的优化，而且使经济市场的准确预测成为可能。最后，我们建议研究的功能理论的特定分支，构成了最近开发的H-赋值控制理论的数学核心，该理论已用于设计Tokamaks内部融合反应的控制系统，并为航天飞机设计了反馈稳定系统。