The Theory of Composition Operators on Analytic Spaces:
Where Has It Been and Where Is It Going?
Carl C. Cowen
Purdue University
Over the past half century, there has been strong interplay between
complex analysis on the one hand and the theory of operators acting on
function spaces on the other. The operators and spaces are seen as
concrete examples in which progress on structure can be made to provide
models for abstract operators on abstract spaces or illuminate
structural questions about them. Often the goal is to relate the
functional analytic behavior of the operators with the geometric or
function theoretic behavior of the functions involved in defining the
operators.
If \Omega is a domain in C or C^N, \phi is an
analytic map of \Omega into itself, and H is a Hilbert (or
Banach) space of functions analytic on \Omega, the composition
operator C_\phi is the linear transformation on H given by
f --> f o \phi for f in H.
Questions that have been studied concerning these operators include
determination of boundedness or compactness, determination of spectrum or
essential spectrum, questions about cyclicity, questions about the
internal structure of the operators, and about connections to normality.
This talk will be an incomplete description of some areas of progress
over the past thirty years or so in the study of these operators,
pointing out some of the successes and some of the areas in which there
is still no substantial success. In addition to pointing out some areas
in which there are intractable problems, there will be an attempt to
suggest some areas in which there are interesting, (probably) solvable
problems and some of the motivations for further study.
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Last Update: October 10, 2001