Extending Hilbert Space Operators

在经典牛顿物理学中，物体的位置和动量被假定为在某个时刻同时可知。现代物理学的一个早期发现是，这种情况在亚原子水平上非常不同。例如，维尔纳·海森堡著名的测不准原理断言，不可能同时测量量子粒子（如绕原子核运行的电子）的位置和动量。 1926年，冯·诺依曼为亚原子粒子的实际测量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。算子论是现代数学中研究算子的分支，在过去的88年里不断发展，成为数学中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开发，以及现有技术的应用，以解决数学中的一些问题。虽然这不是该项目的主要重点，但该研究在数学物理和控制理论方面有许多可能的应用。现代算子理论的一个支柱是Sz。Nagy Dilation Theorem，它通过将作用在希尔伯特空间上的收缩扩展到作用在更大空间上的协等距来模拟收缩。这一定理及其众多的改进打开了大门，研究解析函数在一个和几个变量通过使用运营商理论的方法。主要研究人员将研究各种问题，在几个复杂的变量和其他领域的分析使用这些运营商的理论方法。特别是，他将使用算子理论的方法来研究以下问题：Nevanlinna-Pick和Cartheodory-Fejer型插值问题;定义在多圆盘和多半平面上的解析函数的边界行为; Caratheodory和小林极值问题引起的极值全纯映射的推导和描述理论;和典型的推导表示公式的解析函数在特定的类，如舒尔，Herglotz，皮克，Loewner，Bessmertnii，和Stieljes类。相关的研究重点是应用交换建模方法来发展多个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.