Extending Hilbert Space Operators

扩展希尔伯特空间算子

基本信息

批准号：
1665260
负责人：
Jim Agler
金额：
$ 18.6万
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：
2017
资助国家：
美国
起止时间：
2017-07-01 至 2020-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1665260&HistoricalAwards=false
关键词：
Extending Hilbert Space Operators

项目摘要

In classical Newtonian physics, the location and movement of a body are simultaneously measurable at a moment in time. An early discovery of modern physics was that this state of affairs is very much different at the subatomic level. In particular, the uncertainty principle of quantum mechanics asserts that it is impossible to simultaneously determine the position and momentum of a quantum particle with arbitrary precision. In 1926, von Neumann laid the precise mathematical foundation for what it is that can actually be measured in the case of a subatomic particle. This work involved the use of operators, linear transformations acting on infinite-dimensional Hilbert spaces. Operator theory, the branch of modern mathematics that studies operators, has subsequently developed to become a far-reaching area of research in pure mathematics, and numerous additional applications throughout mathematics, physics, and engineering have been discovered. This research project involves the development of new techniques within operator theory as well as the application of established techniques to attack a number of questions involving the theory of functions. While it is not the primary focus of the project, the research has many potential applications to mathematical physics, control theory, and the theory of optimization. A pillar of modern operator theory is the Sz.-Nagy dilation theorem, which models contractions acting on Hilbert space by extending them to co-isometries acting on larger spaces. This theorem and its numerous refinements open the door to studying holomorphic functions in one and several variables through the use of operator-theoretic methods. This project studies a variety of problems in several complex variables and other areas of analysis using these operator-theoretic methods. In particular, operator-theoretic methods will be employed to study: interpolation problems of Nevanlinna-Pick and Cartheodory-Fejer type; the boundary behavior of analytic functions defined on polydiscs and poly-halfplanes; the derivation and descriptive theory of extremal holomorphic mappings arising from the Caratheodory and Kobayashi extremal problems; and the canonical derivation of representation formulae for analytic functions in specific classes such as the Schur, Herglotz, Pick, Loewner, Bessmertnyi, and Stieltjes classes on domains in several variables. A related focus of the research is to apply modifications of the modeling methods to develop the theory of analytic functions on varieties in commuting variables and on free domains in several non-commuting variables.

在古典牛顿物理学中，身体的位置和运动是可以在时间的时候同时测量的。现代物理学的早期发现是，这种情况在亚原子层面有很大不同。特别是，量子力学的不确定性原理断言，不可能以任意精度同时确定量子粒子的位置和动量。 1926年，冯·诺伊曼（von Neumann）为在亚原子粒子的情况下实际上可以测量的精确数学基础。这项工作涉及使用操作员，作用于无限二维希尔伯特空间的线性变换。运营商理论是研究运营商的现代数学分支，随后发展成为纯数学研究的深远研究领域，并且在整个数学，物理和工程学中都被发现了许多其他应用。该研究项目涉及运营商理论中新技术的发展，以及既定技术来攻击涉及功能理论的许多问题。尽管这不是项目的主要重点，但该研究在数学物理学，控制理论和优化理论中具有许多潜在的应用。现代操作者理论的支柱是SZ.-NAGY扩张定理，该定理通过将它们扩展到对较大空间的共同体来模拟作用于希尔伯特空间的收缩。该定理及其多次改进为通过使用操作者理论方法在一个和几个变量中研究霍明型功能的大门打开了大门。该项目使用这些操作者理论方法研究了几个复杂变量和其他分析领域的各种问题。特别是，将采用操作者理论方法来研究：Nevanlinna-Pick和Cartheodory-Fejer类型的插值问题；分析函数的边界行为在聚盘和多半平面上定义；由Caratheodory和Kobayashi极端问题引起的极端霍明型映射的衍生和描述理论；以及在多个变量中的域上的schur，herglotz，pick，loewner，besmertnyi和stieltjes类的特定类别中的表示函数的表示公式的规范推导。该研究的一个相关重点是应用建模方法的修改，以在通勤变量和几个非交通变量中的自由域中开发分析功能理论。