# 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

## Access Full Article

top## Abstract

top## How to cite

topJaoua, 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

top- [1] J. S. Choa and H. O. Kim, Compact composition operators on the Nevanlinna class, Proc. Amer. Math. Soc. 125 (1997), 145-151. Zbl0860.47018
- [2] P. L. Duren, Theory of ${H}^{p}$ Spaces, Academic Press, 1970. Zbl0215.20203
- [3] P. L. Duren, On the Bloch-Nevanlinna conjecture, Colloq. Math. 20 (1969), 295-297. Zbl0187.02601
- [4] J. B. Garnett, Bounded Analytic Functions, Academic Press, 1981. Zbl0469.30024
- [5] H. Hunziker and H. Jarchow, Composition operators which improve integrability, Math. Nachr. 152 (1991), 83-99. Zbl0760.47015
- [6] H. Jarchow, Some functional analytic properties of composition operators, Quaestiones Math. 18 (1995), 229-256. Zbl0828.47026
- [7] N. N. Lebedev, Special Functions and their Applications, Academy of Sciences, USSR, 1972.
- [8] E. Nordgren, Composition operators, Canad. J. Math. 20 (1968), 442-449. Zbl0161.34703
- [9] J. W. Roberts and M. Stoll, Composition operators on ${F}^{+}$, Studia Math. 57 (1976), 217-228.
- [10] H. J. Schwartz, Composition operators on ${H}^{p}$, thesis, University of Toledo, 1969.
- [11] J. H. Shapiro, The essential norm of a composition operator, Ann. of Math. 125 (1987), 375-404. Zbl0642.47027
- [12] J. H. Shapiro, Composition Operators and Classical Function Theory, Springer, 1993. Zbl0791.30033
- [13] J. H. Shapiro and A. L. Shields, Unusual topological proporties of the Nevanlinna class, Amer. J. Math. 97 (1975), 915-936. Zbl0323.30033
- [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] J. A. Shoat and J. D. Tamarkin, The Problem of Moments, Amer. Math. Soc., 1943.
- [16] N. Yanagihara, Multipliers and linear functionnals for the class ${N}^{+}$, Trans. Amer. Math. Soc. 180 (1973), 449-461.
- [17] N. Yanagihara, Mean growth and Taylor coefficients of some classes of functions, Ann. Polon. Math. 30 (1974), 37-48. Zbl0244.30032
- [18] N. Yanagihara, The containing Fréchet space for the class ${N}^{+}$, Duke Math. J. 40 (1973), 93-103.
- [19] K. Zhu, Operator Theory in Function Spaces, Marcel Dekker, New York, 1990. Zbl0706.47019

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.