Multi-variable Hardy Spaces and Operator Theory

经典的哈代空间的解析函数在复杂的单位磁盘发挥了开创性的作用，在发展的许多分支的纯数学和应用数学，包括算子理论，复变函数理论，信号处理和控制理论。受这些重要应用的启发，我建议将哈代空间理论中的强有力的结果从一个变量扩展到多个变量，并将这个新的理论应用于多变量算子理论，多个复变量，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，完全Nevanlinna-Pick理论，希尔伯特空间收缩的模型理论，以及数学物理中自伴算子的有限秩扰动理论的关键方面都受到经典哈代空间结果的启发和/或发展。很自然地，人们期望（事实也证明了这一点）哈代空间理论的多元扩展对于现代多元算子理论和多元（交换和非交换）解析函数理论的发展同样重要。 单变量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，我利用这一点（与我的合作者）获得了重要经典结果的忠实推广。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其他主要数学分支的联系，同时保留了单变量理论的美丽和基本结构。 我提议组建一个研究小组，成员包括一名理学硕士，一名博士。学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，我的团队将开发哈代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连接和与测度理论的相互作用，以及哈代空间函数的详细因式分解和结构结果。经典上，这些谱理论，扰动理论，控制理论和算子模型理论提供了强大的工具，这些结果的多元扩展将有同样有价值的应用。从事这项研究将为我的团队中高素质的人员提供培训和发展，它将具有广泛的科学吸引力和影响力。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.