Paul Bourdon -- National Science Foundation Three-Year Grant

Title: Collaborative research in operator theory on holomorphic function spaces

ABSTRACT. Professor Bourdon and his collaborator Joel Shapiro of Michigan State University will investigate problems arising from the interaction between the modern theory of linear operators and the classical theory of analytic functions. The problems to be studied involve norms, decomposability, and numerical ranges of composition operators as well as the chaotic behavior of both composition operators and operators commuting with backward shifts. Insights and tools developed during the course of the project will be applied to other classes of linear operators on Hilbert and Banach spaces.

Many of the differential and integral equations that physicists and engineers use to model physical processes may be viewed as linear operators on spaces of functions. This viewpoint, pioneered by David Hilbert, led to the development of function-theoretic operator theory, which is the branch of mathematics inspiring the problems that are the focus of Bourdon and Shapiro's project. A number of these problems concern the notion of numerical range of a linear operator, an object that has relevance to quantum physics and that has proven useful to engineers in determining the stability of certain control systems. Other problems relate to the chaotic behavior of linear operators. That linear operators can give rise to chaotic systems is a relatively recent discovery which has led to unexpected connections between operator theory and dynamical systems. The idea of decomposability--the study of how to break a complicated linear system up into simpler ones--has been shown to have surprising connections with such chaotic behavior. The final group of problems involves norm calculations for composition operators. The norm of an operator measures how much the operator can stretch the unit ball of the space on which it acts. Computer experimentation should yield insights and intuition concerning both composition-operator norms and numerical ranges. Through such experimentation, undergraduate students at Washington and Lee University will be given an opportunity to participate in the project. The training and research experience provided to these students by the project contribute to its human-resources impact.