Topics in Composition Operators

University Of Florida, Gainesville FL

Investigators

Abstract

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.