Multi-variable Hardy Spaces and Operator Theory

多变量Hardy空间和算子理论

• 批准号: RGPIN-2020-05683
• 负责人: Martin, Robert
• 金额: $ 1.31万
• 依托单位: University of Manitoba
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=717920
• 关键词: Multi; variable; Hardy; Spaces; Operator

The classical Hardy spaces of analytic functions in the complex unit disk have played a seminal role in the development of many branches of pure and applied mathematics including Operator Theory, Complex Function Theory, Signal Processing, and Control Theory. Motivated by these important applications, I propose to extend powerful results in Hardy Space Theory from one to several variables, and to apply this new theory to the highly-active fields of Multi-variable Operator Theory, Several Complex Variables, Non-commutative Function Theory, and Operator Algebra Theory. Numerous fundamental concepts in Operator Theory, in particular, were first discovered in the setting of Hardy spaces. For example, the fields of Dilation Theory, Complete Nevanlinna-Pick Theory, model theory for Hilbert space contractions, and key aspects of the theory of finite rank perturbations of self-adjoint operators in Mathematical Physics were all inspired by, and/or developed with classical Hardy space results. It is natural to expect (and this has already proven to be the case) that several-variable extensions of Hardy Space Theory will be similarly important for the advancement of modern Multi-variable Operator Theory and several (commuting and non-commuting) variable analytic function theory. The analogy between single and several-variable Hardy Spaces has proven to be very fruitful, and I have exploited this (with my collaborators) to obtain faithful extensions of significant classical results. Our work has demonstrated that Multi-variable Hardy Space Theory exhibits interesting new phenomena and connections to other major branches of mathematics while retaining the beauty and the essential structure of the single-variable theory. I propose to assemble a research team of one M.Sc. student, one Ph.D. student and two postdoctoral fellows to help achieve this research program. In the next five years my team will develop several-variable analogues of two philosophical cornerstones of Hardy Space Theory: the function-theoretic connection and interplay with Measure Theory on the unit circle, and the detailed factorization and structure results for Hardy space functions. Classically these provide powerful tools for Spectral Theory, Perturbation Theory, Control Theory and Operator Model Theory, and multivariate extensions of these results will have similarly valuable applications. Pursuing this research will provide training and development for the highly qualified personnel on my team, and it will have broad scientific appeal and impact. This is an exciting opportunity to discover new mathematics, and I look forward to realizing this research with students and collaborators.

复杂单元磁盘中分析功能的经典耐寒空间在纯数学和应用数学的许多分支的发展中发挥了开创性的作用，包括操作者理论，复杂函数理论，信号处理和控制理论。在这些重要应用的推动下，我建议将强大的空间理论中的强大结果从一个变量扩展到几个变量，并将这种新理论应用于多变量运算符理论的高度活跃领域，几个复杂的变量，非交换函数理论和操作员代数理论。 尤其是在耐力空间的环境中，尤其是操作者理论中的许多基本概念。例如，扩张理论的领域，完整的Nevanlinna-PICK理论，希尔伯特空间收缩的模型理论以及有限理论在数学物理学中的自我选择算子的扰动的关键方面都受到了经典硬性空间结果的启发和/或开发的。很自然（这已经证明是这种情况）是很自然的，Hardy空间理论的多个可变量扩展对于现代多变量运算符理论的发展和几种（通勤和非交易）的可变分析功能理论同样重要。 事实证明，单个和多变量的耐力空间之间的类比是非常富有成果的，我（与我的合作者）对此进行了利用，以获得忠实的古典成果的忠实扩展。我们的工作表明，多变量的Hardy空间理论表现出有趣的新现象和与数学其他主要分支的联系，同时保留了单变量理论的美丽和基本结构。 我建议组建一个由一个M.Sc的研究团队学生，一位博士学生和两个博士后研究员有助于实现这一研究计划。在接下来的五年中，我的团队将开发出强大空间理论的两个哲学基石的几种可变类似物：功能理论的连接和与单位圈子上的测量理论相互作用，以及Hardy空间功能的详细分解和结构结果。从经典上讲，这些为光谱理论，扰动理论，控制理论和操作者模型理论以及这些结果的多元扩展提供了强大的工具，将具有类似的有价值的应用。进行这项研究将为我团队中高素质的人员提供培训和发展，这将具有广泛的科学吸引力和影响。这是发现新数学的激动人心的机会，我期待与学生和合作者一起实现这项研究。