Extending Hilbert Space Operators

在经典牛顿物理学中，物体的位置和运动在某一时刻是可以同时测量的。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。特别是，量子力学的测不准原理断言，不可能同时以任意精度确定量子粒子的位置和动量。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。这项工作涉及到算子的使用，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，后来发展成为纯数学中一个影响深远的研究领域，并且在数学、物理和工程中发现了许多其他应用。这个研究项目涉及到算子理论中新技术的发展，以及应用已有的技术来解决涉及函数理论的一些问题。虽然这不是该项目的主要焦点，但该研究在数学物理，控制理论和优化理论方面具有许多潜在的应用。现代算符理论的一个支柱是Sz。-纳吉膨胀定理，它通过将作用于希尔伯特空间的收缩扩展到作用于更大空间的共等距来模拟收缩。这个定理及其众多的改进为利用算子理论方法研究单变量和多变量全纯函数打开了大门。本课题利用这些算子理论方法研究了几个复杂变量和其他分析领域的各种问题。特别是，将采用算子理论方法研究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；以及解析函数在特定类(如Schur， Herglotz, Pick, Loewner， Bessmertnyi和Stieltjes类在多个变量域上的表示公式的规范推导。一个相关的研究重点是应用建模方法的修正来发展可交换变量的变异和若干非可交换变量的自由域的解析函数理论。

In classical Newtonian physics, the location and movement of a body are simultaneously measurable at a moment in time. An early discovery of modern physics was that this state of affairs is very much different at the subatomic level. In particular, the uncertainty principle of quantum mechanics asserts that it is impossible to simultaneously determine the position and momentum of a quantum particle with arbitrary precision. 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 work involved the use of operators, linear transformations acting on infinite-dimensional Hilbert spaces. Operator theory, the branch of modern mathematics that studies operators, has subsequently developed to become a far-reaching area of research in pure mathematics, and numerous additional applications throughout mathematics, physics, and engineering have been discovered. This research project involves the development of new techniques within operator theory as well as the application of established techniques to attack a number of questions involving the theory of functions. While it is not the primary focus of the project, the research has many potential applications to mathematical physics, control theory, and the theory of optimization. A pillar of modern operator theory is the Sz.-Nagy dilation theorem, which models contractions acting on Hilbert space by extending them to co-isometries acting on larger spaces. This theorem and its numerous refinements open the door to studying holomorphic functions in one and several variables through the use of operator-theoretic methods. This project studies a variety of problems in several complex variables and other areas of analysis using these operator-theoretic methods. In particular, operator-theoretic methods will be employed to study: interpolation problems of Nevanlinna-Pick and Cartheodory-Fejer type; the boundary behavior of analytic functions defined on polydiscs and poly-halfplanes; the derivation and descriptive theory of extremal holomorphic mappings arising from the Caratheodory and Kobayashi extremal problems; and the canonical derivation of representation formulae for analytic functions in specific classes such as the Schur, Herglotz, Pick, Loewner, Bessmertnyi, and Stieltjes classes on domains in several variables. A related focus of the research is to apply modifications of the modeling methods to develop the theory of analytic functions on varieties in commuting variables and on free domains in several non-commuting variables.