Date of Award

Fall 1996

Project Type

Dissertation

Program or Major

Mathematics

Degree Name

Doctor of Philosophy

First Advisor

Eric A Nordgren

Abstract

General background. Composition operators are defined on a Hilbert (or Banach) spaces of complex valued functions defined on some set X. For the big majority of cases the set X is the unit disc in the complex plane, and the space of functions is one of the Hardy or Bergman spaces (weighted or not). This is due, without doubt, to the richness of those spaces, and the high degree of interest in them. There have been also important papers on the study of the Hardy spaces on the unit ball of n-dimensional complex space.

I have been working with my advisor, Prof. Eric Nordgren, towards generalizing the setting of the composition operators from the spaces above to Hilbert spaces of differential forms on Riemann surfaces. The composition operator then becomes the familiar "pullback", but its properties as an operator between Hilbert spaces have not been taken into consideration before.

Results. We have found a characterization for the boundedness of the composition operator induced by an analytic map on the spaces of measurable square integrable 1-forms on two Riemann surfaces and answered the compactness questions for this case. We also found corresponding characterizations for the operator acting on the spaces of square integrable analytic forms.

We discovered that if we consider the operator on the space of analytic square integrable forms of the unit disc, it is invertible if and only if the inducing map is invertible. This is also the case on all compact surfaces, with the exception of tori. In Chapter 4 we showed that the invertibility of the operator implies that the inducing function is one to one on more general Riemann Surfaces.

Some results were obtained in computing the spectrum of the operator induced by automorphisms of the unit disc. However, a complete description seems hard to get.

In the last chapter we give two examples of composition operators acting on spaces of square integrable analytic 1-forms on open Riemann surfaces.

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