Revista Matemática Iberoamericana


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Volume 28, Issue 1, 2012, pp. 57–76
DOI: 10.4171/RMI/666

Some revisited results about composition operators on Hardy spaces

Pascal Lefèvre [1], Daniel Li[2], Hervé Queffélec[3] and Luis Rodríguez Piazza[4]

(1) Laboratoire de Mathématiques de Lens EA 2462, Université d'Artois, rue Jean Souvraz S.P. 18, 62307, LENS CEDEX, FRANCE
(2) Laboratoire de Mathématiques de Lens EA 2462, Université d'Artois, rue Jean Souvraz S.P. 18, 62307, LENS CEDEX, FRANCE
(3) Laboratoire Paul Painlevé, UMR CNRS 8524, Université Lille I, Bâtiment M2, Cité scientifique, 59655, VILLENEUVE D'ASCQ CEDEX, FRANCE
(4) Departamento de Análisis Matemático, Universidad de Sevilla, Apartado de Correos 1160, 41080, SEVILLA, SPAIN

On the one hand, we generalize 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 $H^\Psi$ and yet “nowhere differentiable”; construction of a surjective symbol whose associated composition operator is compact on~$H^\Psi$ and is, moreover, in all 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 simpler, characterization of composition operators with closed range, and then showing directly the equivalence of the two characterizations of membership in Schatten classes of Luecking, and Luecking–Zhu.

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

Lefèvre Pascal, Li Daniel, Queffélec Hervé, Rodríguez Piazza Luis: Some revisited results about composition operators on Hardy spaces. Rev. Mat. Iberoamericana 28 (2012), 57-76. doi: 10.4171/RMI/666