Extending Hilbert Space Operators

扩展希尔伯特空间算子

• 批准号: 1068830
• 负责人: Jim Agler
• 金额: $ 18.97万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-07-01 至 2014-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1068830&HistoricalAwards=false
• 关键词: Extending; Hilbert; Space; Operators

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 many 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, the theories of functions in one and several complex variables, and the development of holomorphic function theory on analytic varieties. One group of problems that we propose to investigate involve the generalizations to several complex variables of 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 extending the Caratheodory-Julia Theory to several variables with a long term goal of building a geometric approach to the boundary regularity of holomorphic mappings. Research intrinsic to operator theory that we will undertake includes issues involving model theory in one variable on non-simply connected domains in the plane and in several variables on domains other than the bidisc as well as the generalization to several variables of the work of Loewner on operator monotone functions.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 and Quadratic Programming. Linear Matrix Inequalities and Quadratic Programming, are a far reaching extension of Linear Programming, which has made large scale resource allocation and economic prediction possible. In addition, to being a powerful tool in engineering, they have been used to develop state of the art algorithms for global positioning with incomplete sensor location data . Finally, the particular brand 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，很难想象有一个定理对算子理论的发展及其在数学和科学中的许多应用产生了比Sz定理更深刻的影响。Nagy膨胀定理这个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，运营商理论，理论的功能在一个和几个复杂的变量，和发展全纯函数理论的解析品种。一组问题，我们建议调查涉及的推广到几个复杂的变量的经典时刻和插值问题的单位圆盘，如插值定理的Nevanlinna和皮克，矩定理的Caratheodory和Herglotz。另一组涉及扩展Caratheodory-Julia理论的几个变量的长期目标是建立一个几何方法的边界正则性的全纯映射。研究内在的算子理论，我们将承担包括涉及模型理论的问题在一个变量上的非单连通域在平面上，在几个变量上的域以外的bidisc以及推广到几个变量的工作Loewner算子单调函数。算子理论，特定类型的数学，我们建议调查，对人类奋进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和二次规划。线性矩阵不等式和二次规划是线性规划的一个广泛的推广，它使大规模的资源分配和经济预测成为可能。此外，作为一个强大的工具，在工程上，他们已被用来开发最先进的算法，全球定位与不完整的传感器位置数据。最后，我们提出要研究的函数理论的特殊品牌，形成了H ∞控制理论的数学核心，它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。