Extending Hilbert Space Operators

在经典牛顿物理学中，假设物体的位置和动量在某一时刻是同时可知的。现代物理学的一个早期发现是，这种情况在亚原子水平上是非常不同的。例如，维尔纳·海森堡著名的测不准原理断言，不可能同时测量一个量子粒子的位置和动量，比如绕原子核运行的电子。1926年，冯·诺伊曼为亚原子粒子的可测量性奠定了精确的数学基础。这一开创性的突破涉及到“算子”的使用，即作用于无限维希尔伯特空间的线性变换。算子理论是现代数学中研究算子的分支，在过去的88年里，它已经发展成为数学中一个影响深远的研究领域，对数学、物理和工程的许多领域产生了重大影响。该项目涉及算子理论中新技术的发展，以及已建立的技术的应用，以解决数学中的一些问题。虽然这不是该项目的主要焦点，但这项研究在数学物理和控制理论方面都有许多可能的应用。现代算符理论的一个支柱是Sz。-纳吉膨胀定理，它通过将希尔伯特空间上的收缩扩展到更大空间上的共等距来模拟收缩。这个定理和它的许多改进为通过使用算子理论方法研究单变量和多变量解析函数打开了大门。首席研究员将使用这些算子理论方法研究几个复杂变量和其他分析领域的各种问题。特别是，他将使用算子理论方法研究以下问题：Nevanlinna-Pick和Cartheodory-Fejer型插值问题；定义在多盘和半平面上的解析函数的边界行为Caratheodory和Kobayashi极值问题引起的极值全纯映射的推导和描述理论；以及特定类（如Shur、Herglotz、Pick、Loewner、Bessmertnii和Stieljes类）中解析函数表示公式的规范推导。一个相关的研究焦点是应用交换建模方法来发展非交换变量的解析函数理论。

In classical Newtonian physics, the position and momentum of a body are assumed to be simultaneously knowable at an instant in time. An early discovery of modern physics was that this situation is very much different at the subatomic level. For example, the famous Uncertainty Principle of Werner Heisenberg asserts that it is impossible to measure simultaneously the position and momentum of a quantum particle such as an electron orbiting the nucleus of an atom. In 1926, von Neumann laid the precise mathematical foundation for what it is that can actually be measured in the case of a subatomic particle. This seminal breakthrough involved the use of "operators," linear transformations acting on infinite dimensional Hilbert spaces. Operator theory, the branch of modern mathematics that studies operators, has grown over the last eighty-eight years to become a far-reaching area of research in mathematics that has had a major impact on many areas of mathematics, physics, and engineering. This project involves the development of new techniques within operator theory, as well as the application of established techniques, to attack a number of problems within mathematics. Though it is not the major focus of the project, the research has many possible applications to both mathematical physics and control theory. A pillar of modern operator theory is the Sz.-Nagy Dilation Theorem, which models a contraction acting on Hilbert space by extending it to a co-isometry acting on a larger space. This theorem and its numerous refinements open the door to studying analytic functions in one and several variables through the use of operator-theoretic methods. The principal investigator will study a variety of problems in several complex variables and other areas of analysis using these operator-theoretic methods. In particular, he will use operator-theoretic methods to study the following: interpolation problems of Nevanlinna-Pick and Cartheodory-Fejer type; the boundary behavior of analytic functions defined on polydiscs and polyhalfplanes; the derivation and descriptive theory of extremal holomorphic mappings arising from the Caratheodory and Kobayashi extremal problems; and the canonical derivation of representation formulas for analytic functions in specific classes such as the Shur, Herglotz, Pick, Loewner, Bessmertnii, and Stieljes classes. A related focus of the research is to apply the commutative modeling methods to develop the theory of analytic functions in several noncommuting variables.