Special operators on classical spaces of analytic functions

Abstract

In the first part of these notes we survey a number of results on composition operators on Hardy spaces and weighted Bergman spaces on the open unit disc D in C. In a straightforward manner we can identify such operators in the Bergman case as formal identities from the ambient spaces into Lebesgue spaces which are associated with so-called Carleson measures on D and we convince ourselves that the extension to Hardy spaces requires to take into account analogous measures on the closed disc D−−. We discuss measures of this kind along with the resulting embeddings into Lebesgue spaces in some detail, and we show how results known, e.g., for composition operators can be generalized to such embeddings. Carleson measures depending on certain fixed parameters form a Banach lattice whose identification as the dual of an appropriate function space are among the topics of the final section.
These notes are based on published and forthcoming papers by several authors. Their purpose was to serve as classroom notes distributed among the participants of the 2003 summer course in Laredo.