Extending Hilbert Space Operators

扩展希尔伯特空间算子

• 批准号: 9801461
• 负责人: Jim Agler
• 金额: $ 15.57万
• 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: 1998
• 资助国家: 美国
• 起止时间: 1998-07-01 至 2002-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9801461&HistoricalAwards=false
• 关键词: Extending; Hilbert; Space; Operators

Agler. Abs Proposal: DMS-9801461 Principal Investigator: Jim Agler 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 myriad applications to mathematics and science than the Sz.-Nagy Dilation Theorem. The idea of representing a general operator in a specified 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 unsolved problems in matrix theory, operator theory, and the theories of functions in one and several complex variables. A particular group of problems that we propose to attack involve 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 con- nected 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 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. LMIs, which currently are all the rage in several areas of engineering, are an extension of linear programming, which has made large scale resource allocation and economic prediction possible. Finally, the particular brand of functi on 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.

阿格勒ABS 提案：DMS-9801461主要研究者：Jim Agler 摘要： 在谱定理之后，很难想象有一个定理对算子理论的发展及其在数学和科学中的无数应用产生了比Sz定理更深刻的影响。Nagy膨胀定理 的想法表示一个一般的运营商在一个指定的一类运营商的一部分，一个很好的运营商在类已经取得了许多成功，我们寻求发展这一观点的主要重点是未解决的问题，在矩阵理论，算子理论和理论的功能，在一个和几个复杂的变量。一组特殊的问题，我们建议攻击涉及的推广到几个复杂的变量的一些经典的时刻和插值问题的单位盘，如插值定理的Nevanlinna和挑选，和矩定理的Caratheodory和Herglotz。另一组涉及推导类似的定理阿达米扬，Arov，和克莱因的空间更一般比经典的哈代空间。研究内在的算子理论，我们将进行包括涉及模型理论的问题，在一个变量的非简单连接域的平面和几个变量的域比bidisc。 算子理论是我们打算研究的一种特殊类型的数学，它对人类努力的许多领域都有直接和具体的好处。 例如，我们的建议的模型理论方面都涉及到推广的换向提升结构，这导致了一个有效的算法，发现石油从声学数据在地球表面上。 其他方面将添加到理论的线性矩阵不等式。 线性矩阵不等式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。最后，我们建议研究的特定品牌的函数理论形成了最近发展的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。