Multi-variable Hardy Spaces and Operator Theory

复单位盘上解析函数的经典Hardy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，包括算子论、复函数论、信号处理和控制论。在这些重要应用的激励下，我建议将Hardy空间理论的强大结果从一个变量扩展到几个变量，并将这一新理论应用于多变量算子理论、几个复变量、非交换函数理论和算子代数理论等高度活跃的领域。特别是算子理论中的许多基本概念，都是在Hardy空间中首次发现的。例如，膨胀理论、完全奈万林纳-匹克理论、希尔伯特空间收缩模型理论，以及数学物理中自伴随算子有限秩摄动理论的关键方面，都是受到经典Hardy空间结果的启发，或者是在经典Hardy空间结果的基础上发展起来的。我们很自然地期望（这已经被证明是事实）哈代空间理论的几个变量扩展对于现代多变量算子理论和几个（交换和非交换）变量解析函数理论的发展同样重要。单变量和多变量Hardy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和与其他主要数学分支的联系，同时保留了单变量理论的美和基本结构。我建议组建一个由一名硕士研究生、一名博士研究生和两名博士后组成的研究团队来帮助完成这个研究项目。在接下来的五年里，我的团队将开发Hardy空间理论的两个哲学基石的几个变量类似物：单位圆上与测度理论的泛函联系和相互作用，以及Hardy空间函数的详细分解和结构结果。这些结果为谱理论、摄动理论、控制理论和算子模型理论提供了有力的工具，这些结果的多变量扩展将具有类似的有价值的应用。从事这项研究将为我的团队中高素质的人才提供培训和发展，并将具有广泛的科学吸引力和影响。这是一个令人兴奋的发现新数学的机会，我期待着与学生和合作者一起实现这项研究。

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.