Topics in Composition Operators

组合运算符主题

• 批准号: 0701268
• 负责人: Michael Jury
• 金额: $ 9.68万
• 依托单位: University of Florida
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-06-01 至 2011-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0701268&HistoricalAwards=false
• 关键词: Topics; Composition; Operators

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.

这个项目将研究现代泛函分析和算子理论与经典复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算子和复合算子之间的代数关系。将探索单位圆子集上的动力系统与复合算符之间的一种新的联系，一种基于与量子力学系统动力学平行的联系。首席研究人员希望开发创新的技术，在单变量和多变量环境下计算范数，并将现有的与紧致性问题有关的方法应用于有关伴随和代数关系的问题。此外，还将研究最近发现的复合算子和多维线性系统之间的联系。主要研究人员打算使用的方法吸收了现代分析中的广泛技术，包括算子理论、泛函分析、测度论和调和分析。物理和工程中的许多数学问题(以及在纯数学中也是如此)可以表示为关于函数空间上的算子的问题。(这种观点导致了算符理论函数论领域的出现，它起源于20世纪初大卫·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”。许多符合这一标题的问题(例如设计自动驾驶仪以稳定飞机)自然被认为是操作员理论中的问题。这种方法的成功扩大了工程师现在希望解决的问题的类别，包括将一个维度的已知结果扩展到多个维度。在其他方面，首席研究员将研究这些问题与算子理论中的其他问题之间的新联系。