Order bounded composition operators on the Hardy spaces and the Nevanlinna class

Nizar Jaoua

Studia Mathematica, Tome 134 (1999) no. 1, pp. 35-55

Résumé

We study the order boundedness of composition operators induced by holomorphic self-maps of the open unit disc D. We consider these operators first on the Hardy spaces $H^p$ 0 p ∞ and then on the Nevanlinna class N. Given a non-negative increasing function h on [0,∞[, a composition operator is said to be X,L_h-order bounded (we write (X,L_h)-ob) with $X = H^p$ or X = N if its composition with the map f ↦ f*, where f* denotes the radial limit of f, is order bounded from X into $L_h$. We give a complete characterization and a family of examples in both cases. On the other hand, we show that the ($N,log^{+}L$)-ob composition operators are exactly those which are Hilbert-Schmidt on $H^2$. We also prove that the ($N,L^q$)-ob composition operators are exactly those which are compact from N into $H^q$.

Détail

DOI : 10.4064/sm-134-1-35-55

Keywords: composition operators, order bounded maps, Hardy spaces, Nevanlinna class, radial limit, moment sequences and analytic moment sequences