Carleson measures: an efficient tool for compactness or membership in Schatten classes of composition operators
by
Hervé Queffelec
University of Lille

Sophisticated criteria for compactness or membership in Schatten classes of composition operators, due to Mc Cluer, J.Shapiro, Luecking, Zhu, and al. have been known for more than 20 years, either in terms of Nevanlinna counting functions, or in terms of Carleson measures. But these criteria seemed very difficult to check on specific examples. It has been recently realized that:

1) A strenghtened notion of Carleson measure is useful in that context.

2) The Carleson measure attached to a "symbol" (an analytic self-map of the unit disk) can be analyzed accurately for classes of symbols.

This allows:

1) Simplified proofs of results due to Shapiro, Carroll, Cowen, and al. in the nineties.

2) New results in that context, in particular the comparison of compactness of composition operators acting on Hardy spaces on the one-hand , and Hardy-Orlicz spaces on the other hand.

Date received: April 20, 2008