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.

在光谱定理之后，很难想到一个定理对操作者理论的发展及其在数学和科学上的应用比SZ.-nagy膨胀定理具有更深远的影响。在指定类的运营商中代表普通操作员作为班级的好操作员的一部分的想法取得了许多成功，我们试图发展这种观点，主要关注矩阵理论，运算符理论，一个和几个复杂变量的函数理论以及对分析品种的全体形态函数理论的发展。我们建议进行研究的一组问题涉及对单位光盘上的几个复杂变量和插值问题的概括，例如Nevanlinna和Pick的插值定理，以及Caratheodory和Herglotz定理的时刻。另一组涉及将caratheodory-julia理论扩展到几个变量，其长期目标是建立一种几何方法，以实现全体形态映射的边界规则性。 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.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，该结构导致有效的算法从地球表面上获取的声学数据发现油。其他方面将增加线性矩阵不平等和二次编程的理论。线性矩阵不平等和二次编程是线性编程的远程扩展，这使得大规模的资源分配和经济预测成为可能。此外，作为工程技术的强大工具，它们已被用来开发具有不完整的传感器位置数据的全球定位的最先进算法。最后，我们建议研究的功能理论的特定品牌构成了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。