Extending Hilbert Space Operators

扩展希尔伯特空间算子

• 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

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.

在经典牛顿物理学中，物体的位置和运动在某个时刻是可以同时测量的。现代物理学的一个早期发现是，这种状态在亚原子水平上非常不同。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·诺依曼为亚原子粒子的实际测量奠定了精确的数学基础。这项工作涉及使用运营商，线性变换作用于无限维希尔伯特空间。算子理论是研究算子的现代数学的分支，后来发展成为纯数学中一个影响深远的研究领域，并在数学、物理和工程中发现了许多其他应用。本研究项目涉及算子理论中新技术的开发以及既定技术的应用，以解决涉及函数理论的一些问题。虽然这不是该项目的主要重点，但该研究在数学物理，控制理论和优化理论方面有许多潜在的应用。现代算子理论的一个支柱是Sz。Nagy膨胀定理，它通过将作用于希尔伯特空间的收缩扩展到作用于更大空间的余等距来模拟收缩。这个定理和它的许多改进打开了大门，研究全纯函数在一个和几个变量，通过使用算子理论的方法。该项目研究了多个复变量和使用这些算子理论方法的其他分析领域中的各种问题。特别是，算子理论方法将被用来研究：Nevanlinna-Pick和Cartheodory-Fejer型插值问题;定义在多圆盘和多半平面上的解析函数的边界行为; Caratheodory和小林极值问题引起的极值全纯映射的推导和描述理论;和典型的推导表示公式的解析函数在特定的类，如舒尔，Herglotz，皮克，Loewner，Bessmertnyi，和Stieltjes类域在几个变量。一个相关的研究重点是应用修改的建模方法，以发展理论的解析函数的各种交换变量和自由域中的几个非交换变量。

项目成果

Caratheodory extremal functions on the symmetrized bidisc

对称 bidisc 上的 Caratheodory 极值函数

• DOI: 10.1007/978-3-030-04269-1_1
• 发表时间: 2018
• 期刊: Operator Theory Advances and Applications
• 作者: Agler, J.;Lycova, Zinaida A.;Young, N. J.
• 通讯作者: Young, N. J.

Wandering Montel theorems for Hilbert space valued holomorphic functions

希尔伯特空间值全纯函数的徘徊蒙特尔定理

• DOI: 10.1090/proc/14086
• 发表时间: 2018
• 期刊: Proceedings of the American Mathematical Society
• 影响因子: 1
• 作者: Agler, Jim;McCarthy, John E.
• 通讯作者: McCarthy, John E.

A geometric characterization of the symmetrized bidisc

对称bidisc的几何特征

• DOI: 10.1016/j.jmaa.2019.01.027
• 发表时间: 2019
• 期刊: Journal of Mathematical Analysis and Applications
• 影响因子: 1.3
• 作者: Agler, Jim;Lykova, Zinaida;Young, N.J.
• 通讯作者: Young, N.J.

Global Holomorphic Functions in Several Non-Commuting Variables II

多个非交换变量的全局全纯函数 II

• DOI: 10.4153/cmb-2017-044-4
• 发表时间: 2018
• 期刊: Canadian Mathematical Bulletin
• 作者: Agler, Jim;McCarthy, John
• 通讯作者: McCarthy, John