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Multivariable Operator Theory: The Interplay between Function Theory, Operator Theory and Operator Algebras

多变量算子理论：函数论、算子理论和算子代数之间的相互作用

基本信息

批准号：
1900364
负责人：
Michael Jury
金额：
$ 13.44万
依托单位：
University of Florida
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-07-01 至 2023-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1900364&HistoricalAwards=false
关键词：
Multivariable Operator Theory Interplay between

项目摘要

This project belongs to the branch of mathematical analysis known as operator theory. The subject was developed initially as part of the development of the mathematical foundations of quantum mechanics. Since then it has evolved in many directions, with applications not only in physics (such as the currently very active areas of quantum computing and quantum information theory), but also in electrical and mechanical engineering (where its ideas find application in e.g. the design of control systems, or in signal and image processing). A particularly new and exciting branch of this field is known as "noncommutative function theory," which has its origins in the study of certain kinds of optimization problems in engineering. It has now taken on a life of its own. The area of "multivariable operator theory" is multi-faceted but is closely connected with many questions arising in these applications (such as the study of quantum channels, and the theory of "linear matrix inequalities" in optimization). This project is aimed at expanding the array of mathematical tools available for the study of these problems. The goal of this project is to study the connection between operator theory and function theory in several variables, by analogy with the very successful one-variable theory, particularly in the rapidly developing area of "noncommutative function theory." The role of analytic function theory in the one-variable setting is very well established; e.g. a central role is played by the Cauchy transform in the unitary/circle setting or the Borel transform in the selfadjoint/real line setting. Noncommutative functions, or nc-functions for short, are meant to play a role analogous to analytic functions in the classical one-variable theory. They arise, for example, as noncommutative Cauchy transforms of "noncommutative" measures, viewed as states on a C*-algebra or related objects (such as an operator system). Problems to be studied include invariant subspaces for noncommutative operator tuples, in particular an nc-function theoretic analysis of "nc-inner" functions, and the development of a version of "noncommutative measure theory" suitable to the desired applications, in particular a version of the Lebesgue decomposition suited to the analysis, for example, of noncommutative analogs of the Cauchy transform. The project will draw on techniques from several currently active areas of mathematical analysis, with the aim of broadening and deepening understanding of the interplay between function theory, operator theory, and operator algebras. The project will also employ novel methods to study some classical objects of analysis to explore open questions in complex analysis and operator theory.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

这个项目属于被称为算子理论的数学分析分支。这门学科最初是作为量子力学数学基础发展的一部分而发展起来的。从那时起，它在许多方向上发展，不仅应用于物理学（例如目前非常活跃的量子计算和量子信息理论领域），而且应用于电气和机械工程（其思想在控制系统设计或信号和图像处理中得到应用）。这个领域的一个特别新颖和令人兴奋的分支被称为“非交换函数理论”，它起源于对工程中某些类型的优化问题的研究。它现在有了自己的生命。“多变量算子理论”的领域是多方面的，但与这些应用中出现的许多问题密切相关（例如量子通道的研究，以及优化中的“线性矩阵不等式”理论）。这个项目旨在扩大可用于研究这些问题的数学工具的范围。本项目的目标是通过类比非常成功的单变量理论，特别是在迅速发展的“非交换函数理论”领域，研究数变量算子理论和函数理论之间的联系。解析函数理论在单变量情况下的作用已经很好地确立了；例如，在酉/圆的情况下，柯西变换或在自伴/实线的情况下，Borel变换起着中心作用。非交换函数，简称nc函数，在经典单变量理论中扮演着类似解析函数的角色。例如，它们作为“非交换”测度的非交换柯西变换出现，被视为C*代数或相关对象（如算子系统）上的状态。要研究的问题包括非交换算子元组的不变子空间，特别是“nc内”函数的nc函数理论分析，以及适合于期望应用的“非交换测度理论”的发展，特别是适合于分析的Lebesgue分解的版本，例如，柯西变换的非交换类似物。该项目将利用几个当前活跃的数学分析领域的技术，目的是扩大和深化对函数理论、算子理论和算子代数之间相互作用的理解。本项目也将采用新颖的方法来研究一些经典的分析对象，以探索复杂分析和算子理论中的开放性问题。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

期刊论文数量（11）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Unbounded Multipliers of Complete Pick Spaces

完整选择空间的无限乘数

DOI：
10.1007/s00020-022-02690-8
发表时间：
2022
期刊：
Integral Equations and Operator Theory
影响因子：
0.8
作者：
Jury, Michael T.;Martin, Robert T.
通讯作者：
Martin, Robert T.

Non-commutative rational functions in the full Fock space

DOI：
10.1090/tran/8418
发表时间：
2020-10
期刊：
Transactions of the American Mathematical Society
影响因子：
1.3
作者：
M. Jury;R. Martin;E. Shamovich
通讯作者：
M. Jury;R. Martin;E. Shamovich

Noncommutative Partial Convexity Via $$\Gamma $$-Convexity

通过 $$Gamma $$-凸性实现非交换部分凸性

DOI：
10.1007/s12220-020-00387-1
发表时间：
2021
期刊：
The Journal of Geometric Analysis
影响因子：
0
作者：
Jury, Michael;Klep, Igor;Mancuso, Mark E.;McCullough, Scott;Pascoe, James Eldred
通讯作者：
Pascoe, James Eldred

Noncommutative rational Clark measures

非交换理性克拉克测度