Some revisited results about composition operators on Hardy spaces

Pascal Lefèvre (1) , Daniel Li (1) , Hervé Queffélec (2) , Luis Rodriguez-Piazza (3)

1 LML - Laboratoire de Mathématiques de Lens
2 LPP - Laboratoire Paul Painlevé - UMR 8524
3 Departamento de Analisis Matematico

We generalize, on one hand, some results known for composition operators on Hardy spaces to the case of Hardy-Orlicz spaces $H^\Psi$: construction of a ``slow'' Blaschke product giving a non-compact composition operator on $H^\Psi$; construction of a surjective symbol whose composition operator is compact on $H^\Psi$ and, moreover, is in all the Schatten classes $S_p (H^2)$, $p > 0$. On the other hand, we revisit the classical case of composition operators on $H^2$, giving first a new, and simplier, characterization of closed range composition operators, and then showing directly the equivalence of the two characterizations of membership in the Schatten classes of Luecking and Luecking and Zhu.

Keywords: Blaschke product; Carleson function; Carleson measure; composition operator; Hardy-Orlicz space; Nevanlinna counting function; Schatten classes

Domains: Mathematics [math]; Functional Analysis [math.FA]

Pascal Lefèvre, Daniel Li, Hervé Queffélec, Luis Rodriguez-Piazza. Some revisited results about composition operators on Hardy spaces. 2010. ⟨hal-00448623⟩