Composition operators on Hardy spaces

摘要

The topic of this short survey is composition operators, i.e., maps of the shape f→fo Φp, where the symbol Φ : D → D is holomorphic. We shall give a (non-exhaustive) overview of -more or less recent- results on such operators, acting on Hardy spaces. The story involves some classical tools of complex analysis, as Nevanlinna counting function and Carleson measures. We shall illustrate this presentation with miscellaneous examples and questions. We shall pay attention to their possible membership to various classes of operator ideals, and in particular, in the membership of the class of absolutely summing operators.