Multi-variable Hardy Spaces and Operator Theory

复单位圆盘上解析函数的经典Hardy空间对许多纯数学和应用数学的发展起到了开创性的作用，包括算子论、复函数论、信号处理和控制论。受这些重要应用的启发，我建议将Hardy空间理论中的强大结果从一个变量推广到多个变量，并将这一新理论应用到多变量算子理论、多复变量理论、非对易函数理论和算子代数理论等高度活跃的领域。 特别是算子论中的许多基本概念，首先是在Hardy空间的背景下发现的。例如，数学物理中的膨胀理论、完备的Nevanlinna-Pick理论、Hilbert空间压缩的模型理论以及自伴算子的有限秩扰动理论的关键方面都受到经典Hardy空间结果的启发和/或发展。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(可交换和非可交换)解析函数理论的发展同样重要，这是很自然的(已经证明是这样的)。 单变量和多变量Hardy空间之间的类比已被证明是非常有成果的，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Hardy空间理论展示了有趣的新现象，并与其他主要数学分支建立了联系，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理科硕士组成的研究小组。学生，一名博士生和两名博士后研究员，以帮助实现这一研究计划。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：单位圆上的函数理论联系和与测度论的相互作用，以及Hardy空间函数的详细因式分解和结构结果。经典地说，这些结果为谱理论、微扰理论、控制理论和算子模型理论提供了强大的工具，这些结果的多元推广也将有类似的有价值的应用。开展这项研究将为我的团队中的高素质人才提供培训和发展，具有广泛的科学号召力和影响。这是一个发现新数学的令人兴奋的机会，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.