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

Studia Mathematica (1999)

• Volume: 134, Issue: 1, page 35-55
• ISSN: 0039-3223

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## Abstract

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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,Lh-order bounded (we write (X,Lh)-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 ($N,lo{g}^{+}L$)-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 ${H}^{q}$.

## How to cite

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Jaoua, Nizar. "Order bounded composition operators on the Hardy spaces and the Nevanlinna class." Studia Mathematica 134.1 (1999): 35-55. <http://eudml.org/doc/216621>.

@article{Jaoua1999,
abstract = {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,Lh-order bounded (we write (X,Lh)-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 $H^q$.},
author = {Jaoua, Nizar},
journal = {Studia Mathematica},
keywords = {composition operators; order bounded maps; Hardy spaces; Nevanlinna class; radial limit; moment sequences and analytic moment sequences; composition operator; boundedness},
language = {eng},
number = {1},
pages = {35-55},
title = {Order bounded composition operators on the Hardy spaces and the Nevanlinna class},
url = {http://eudml.org/doc/216621},
volume = {134},
year = {1999},
}

TY - JOUR
AU - Jaoua, Nizar
TI - Order bounded composition operators on the Hardy spaces and the Nevanlinna class
JO - Studia Mathematica
PY - 1999
VL - 134
IS - 1
SP - 35
EP - 55
AB - 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,Lh-order bounded (we write (X,Lh)-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 $H^q$.
LA - eng
KW - composition operators; order bounded maps; Hardy spaces; Nevanlinna class; radial limit; moment sequences and analytic moment sequences; composition operator; boundedness
UR - http://eudml.org/doc/216621
ER -

## References

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14. [14] J. H. Shapiro and P. D. Taylor, Compact, nuclear and Hilbert-Schmidt composition operators on ${H}^{2}$, Indiana Univ. Math. J. 125 (1973), 471-496.
15. [15] J. A. Shoat and J. D. Tamarkin, The Problem of Moments, Amer. Math. Soc., 1943.
16. [16] N. Yanagihara, Multipliers and linear functionnals for the class ${N}^{+}$, Trans. Amer. Math. Soc. 180 (1973), 449-461.
17. [17] N. Yanagihara, Mean growth and Taylor coefficients of some classes of functions, Ann. Polon. Math. 30 (1974), 37-48. Zbl0244.30032
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19. [19] K. Zhu, Operator Theory in Function Spaces, Marcel Dekker, New York, 1990. Zbl0706.47019

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