Extending Hilbert Space Operators

在经典牛顿物理学中，物体的位置和运动在某个时刻是可以同时测量的。现代物理学的一个早期发现是，在亚原子水平上，这种情况非常不同。特别是，量子力学的测不准原理断言，不可能以任意精度同时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实际测量奠定了精确的数学基础。这项工作涉及到算子的使用，即作用于无限维希尔伯特空间的线性变换。算符理论是现代数学中研究算符的分支，后来发展成为纯数学中一个影响深远的研究领域，并在数学、物理和工程学中发现了许多其他应用。这项研究项目涉及开发算子理论中的新技术，以及应用已有的技术来解决涉及函数论的一些问题。虽然这不是该项目的主要关注点，但该研究在数学物理、控制理论和最优化理论方面有许多潜在的应用。现代算子理论的一个支柱是Sz.-Nagy膨胀定理，它通过将作用于Hilbert空间的压缩扩展到作用于较大空间的余测来模拟作用于Hilbert空间的压缩。这个定理及其众多的改进为使用算子论方法研究一元多元全纯函数打开了大门。这个项目使用这些算子论方法研究了几个复变量和其他分析领域中的各种问题。特别地，我们将用算子论的方法来研究：Nevanlinna-Pick和Cartheodory-Fejer型的插值问题；定义在多面体和多半平面上的解析函数的边界行为；由Caratheodory和Kobayashi极值问题产生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中的表示公式的典范推导，例如关于多变量区域上的Schur、Hergltz、Pick、Loewner、Bessmertnyi和Stieltjes类。研究的一个相关重点是应用模型方法的改进来发展关于可交换变量中的变种的解析函数理论和关于多个非可交换变量中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.