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: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758547
• 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.

复数单位圆盘中解析函数的经典哈代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，包括算子理论、复函数理论、信号处理和控制理论。受这些重要应用的推动，我建议将哈代空间理论的强大结果从一个变量扩展到多个变量，并将这一新理论应用到多变量算子理论、多复变量、非交换函数论等高度活跃的领域，和算子代数理论。 特别是算子理论中的许多基本概念都是在哈代空间的背景下首次发现的。例如，膨胀理论、完整的 Nevanlinna-Pick 理论、希尔伯特空间收缩的模型理论以及数学物理学中自伴算子的有限阶摄动理论的关键方面都受到了启发和/或发展起来。经典哈代空间结果。很自然地预期（事实已经证明了这一点）哈代空间理论的多变量扩展对于现代多变量算子理论和多变量（通勤和非通勤）变量分析的进步同样重要。函数理论。单变量哈代空间和多变量哈代空间之间的类比已被证明是非常富有成效的，我（与我的合作者）利用这一点来获得重要经典结果的忠实扩展。我们的工作表明，多变量哈代空间理论展示了有趣的新现象以及与其他主要数学分支的联系，同时保留了单变量理论的美感和基本结构。 我建议组建一个由一名硕士组成的研究团队。学生，一名博士学生和两名博士后研究员帮助实现这一研究计划。在接下来的五年中，我的团队将开发哈代空间理论的两个哲学基石的多变量类似物：单位圆上函数理论的联系和与测度论的相互作用，以及哈代空间函数的详细因式分解和结构结果。传统上，这些为谱理论、微扰理论、控制理论和算子模型理论提供了强大的工具，并且这些结果的多元扩展将具有类似的有价值的应用。开展这项研究将为我团队中的高素质人才提供培训和发展，并将产生广泛的科学吸引力和影响。这是一个发现新数学的令人兴奋的机会，我期待与学生和合作者一起实现这项研究。