Topics in Composition Operators

本项目将研究现代泛函分析和算子理论与经典复变函数理论和动力系统相互作用所产生的问题。所要研究的问题涉及复合算子的范数和伴随以及Toeplitz算子与复合算子之间的代数关系。 一个新的连接之间的动力系统的子集的单位圆和复合经营者，一个连接的基础上平行的量子力学系统的动力学，将被探讨。首席研究员希望开发创新的技术，在一个和几个变量的设置规范的计算，并将现有的方法相关的紧度问题的问题，有关伴随和代数关系。此外，最近发现的组合算子和多维线性系统之间的联系将进行调查。主要研究者打算使用的方法借鉴了现代分析中的广泛技术，包括算子理论，泛函分析，测度理论和调和分析。物理和工程（以及纯数学）中的许多数学问题都可以表示为函数空间上的算子问题。(This这一观点导致了算子论函数理论领域的出现，它起源于世纪早期大卫希尔伯特的工作。）这种现象的一个例子是工程中的“反馈控制”。在这个标题下的一系列问题（如设计一个自动驾驶仪来稳定飞机）很自然地成为算子理论中的问题。 这种方法的成功扩展了工程师现在希望解决的问题的类别，包括将一维已知的结果扩展到多个维度。除其他外，首席研究员将研究此类问题与算子理论中其他问题之间的新联系。

This project will study problems arising from the interaction of modern functional analysis and operator theory with classical complex function theory and dynamical systems. The problems to be studied involve norms and adjoints of composition operators and algebraic relations between Toeplitz operators and composition operators. A new connection between dynamical systems on subsets of the unit circle and composition operators, a connection based on parallels with the dynamics of quantum mechanical systems, will be explored. The principal investigator expects to develop innovative techniques for the computation of norms in both the one- and several-variable settings and to apply existing methods related to compactness questions to problems concerning adjoints and algebraic relations. In addition, a recently discovered linkage between composition operators and multidimensional linear systems will be investigated. The methods that the principal investigator intends to use draw on a wide range of techniques in modern analysis, including operator theory, functional analysis, measure theory, and harmonic analysis.Many mathematical problems in physics and engineering (and in pure mathematics, as well) can be expressed as problems about operators on spaces of functions. (This point of view led to the emergence of the field of operator-theoretic function theory, which has its origins in the work of David Hilbert in the early twentieth century.) One example of this phenomenon is "feedback control" in engineering. A host of problems that fit under this heading (such as designing an autopilot to stabilize an aircraft) are naturally posed as problems in operator theory. The success of this approach has expanded the class of problems that engineers now wish to solve, including the extension to several dimensions of results that are known in one dimension. Among other things, the principal investigator will study new connections between such problems and other problems in operator theory.