Collaborative Research in Operator Theory on Holomorphic Function Spaces

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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.