Collaborative Research in Operator Theory on Holomorphic Function Spaces

摘要。波登教授和他的合作者密歇根州立大学的乔尔·夏皮罗将研究现代线性算子理论和经典解析函数理论之间的相互作用所产生的问题。本文研究的问题包括复合算子的形、可分解性和数值范围，以及复合算子和反向交换算子的混沌行为。在项目过程中开发的见解和工具将应用于希尔伯特和巴拿赫空间中的其他类别的线性算子。物理学家和工程师用来模拟物理过程的许多微分方程和积分方程可以看作是函数空间上的线性算子。这种观点由大卫·希尔伯特（DavidHilbert）首创，导致了函数算子理论的发展，这是数学的一个分支，启发了波登和夏皮罗项目关注的问题。其中许多问题涉及线性算子的数值范围的概念，这是一个与量子物理学相关的对象，并且已被证明对工程师确定某些控制系统的稳定性很有用。其他问题与线性算子的混沌行为有关。线性算子可以产生混沌系统是一个相对较新的发现，它导致了算子理论和动态系统之间意想不到的联系。可分解性的概念——研究如何将复杂的线性系统分解成更简单的线性系统——已被证明与这种混沌行为有着惊人的联系。最后一组问题涉及组合运算符的范数计算。一个算子的范数测量了算子能在多大程度上拉伸它所作用的空间的单位球。计算机实验应该产生关于组合算子规范和数值范围的见解和直觉。通过这样的实验，华盛顿和李大学的本科生将有机会参与这个项目。该项目为这些学生提供的培训和研究经验有助于其人力资源影响。

ABSTRACT. Professor Bourdon and his collaborator Joel Shapiro ofMichigan State University will investigate problems arising from theinteraction between the modern theory of linear operators and theclassical theory of analytic functions. The problems to be studied involvenorms, decomposability, and numerical ranges of composition operators aswell as the chaotic behavior of both composition operators and operatorscommuting with backward shifts. Insights and tools developed during thecourse of the project will be applied to other classes of linear operatorson Hilbert and Banach spaces. Many of the differential and integral equations that physicists andengineers use to model physical processes may be viewed as linearoperators on spaces of functions. This viewpoint, pioneered by DavidHilbert, led to the development of function-theoretic operator theory,which is the branch of mathematics inspiring the problems that are thefocus of Bourdon and Shapiro's project. A number of these problems concernthe notion of numerical range of a linear operator, an object that hasrelevance to quantum physics and that has proven useful to engineers indetermining the stability of certain control systems. Other problemsrelate to the chaotic behavior of linear operators. That linear operatorscan give rise to chaotic systems is a relatively recent discovery which has led to unexpected connections between operator theory and dynamicalsystems. The idea of decomposability--the study of how to break acomplicated linear system up into simpler ones--has been shown to havesurprising connections with such chaotic behavior. The final group ofproblems involves norm calculations for composition operators. The normof an operator measures how much the operator can stretch the unit ball ofthe space on which it acts. Computer experimentation should yieldinsights and intuition concerning both composition-operator norms andnumerical ranges. Through such experimentation, undergraduate students atWashington and Lee University will be given an opportunity to participatein the project. The training and research experience provided to thesestudents by the project contribute to its human-resources impact.