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1999 | 134 | 1 | 35-55
Tytuł artykułu

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

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Abstrakty
EN
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 $H^q$.
Czasopismo
Rocznik
Tom
134
Numer
1
Strony
35-55
Opis fizyczny
Daty
wydano
1999
otrzymano
1998-01-19
poprawiono
1998-09-28
Twórcy
autor
Bibliografia
  • [1] J. S. Choa and H. O. Kim, Compact composition operators on the Nevanlinna class, Proc. Amer. Math. Soc. 125 (1997), 145-151.
  • [2] P. L. Duren, Theory of $H^p$ Spaces, Academic Press, 1970.
  • [3] P. L. Duren, On the Bloch-Nevanlinna conjecture, Colloq. Math. 20 (1969), 295-297.
  • [4] J. B. Garnett, Bounded Analytic Functions, Academic Press, 1981.
  • [5] H. Hunziker and H. Jarchow, Composition operators which improve integrability, Math. Nachr. 152 (1991), 83-99.
  • [6] H. Jarchow, Some functional analytic properties of composition operators, Quaestiones Math. 18 (1995), 229-256.
  • [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.
  • [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.
  • [12] J. H. Shapiro, Composition Operators and Classical Function Theory, Springer, 1993.
  • [13] J. H. Shapiro and A. L. Shields, Unusual topological proporties of the Nevanlinna class, Amer. J. Math. 97 (1975), 915-936.
  • [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.
  • [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.
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Bibliografia
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bwmeta1.element.bwnjournal-article-smv134i1p35bwm
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