COMPOSITION OPERATORS ON RIEMANN SURFACES

"Composition Operators on Riemann Surfaces - preliminary results", paper presented at the Canadian Operator Theory Symposium, June 9-13, 1995, Waterloo, ON, Canada.

"Composition Operators on Riemann Surfaces", paper presented at the Composition Operators on Spaces of Analytic Functions Summer Conference organized by the Rocky Mountains Mathematics Consortium, July 8-19 1996, Laramie, WY.

"Boundedness and Compactness of Composition Operators on Riemann Surfaces", paper submitted for publication in the Contemporary Mathematics Monograph series published by AMS.

General background. The composition of functions and the idea of substitution are part of the foundation of mathematics, therefore we can say that composition operators arise naturally. However, their study as a part of operator theory started in the 1960's with the independent papers of E. A. Nordgren and J. V. Ryff, although a fundamental result goes back to Littlewood in 1925. Since then a great amount of work has been done in this field. A survey of the results was published in 1990 by C. C. Cowen in Proceedings of Symposia in Pure Mathematics : "Composition Operators on Hilbert Spaces of Analytic Functions: A Status Report". Besides numerous articles written since the survey, it is worth mentioning the basic book by J. H. Shapiro "Composition Operators and Classical Function Theory", published in 1994.

All these results have been obtained in the following setting: the composition operators are defined on a Hilbert (or Banach) space 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. Nordgren towards generalizing the setting of the composition operators from the spaces above to Hilbert spaces on Riemann surfaces. It was apparent from the first steps taken that spaces of functions will not be appropriate, and that spaces of forms should be used instead. The composition operator then becomes the familiar "pullback", but its properties as an operator between Hilbert spaces have not been taken into consideration before. Questions like boundedness and compactness for various maps of the Riemann surfaces that generate the operator will have to be answered.

Results. We have found so far 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 space of square integrable analytic 1-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 - a rather surprising fact if we look at the difference between our operator and the classical composition operator on the space of analytic functions of the disc.

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.

Questions. Here is a list of related questions that need answers:

• Can these operators be completely characterized in the case of analytic forms on compact Riemann surfaces? (The answer is known if the operator is induced by an automorphism).
• In the case of mapping from a surface into itself, what can be said about the spectrum of the operator? We hope that something can be said for surfaces more complicated than the unit disc.
• How rich are the spaces of analytic square integrable forms on various Riemann surfaces?