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Composition operators on Qp spaces

Part of: Special classes of linear operators Linear function spaces and their duals

Published online by Cambridge University Press:  09 April 2009

Zengjian Lou
Zengjian Lou
Affiliation:
Centre for Mathematics and its Applications School of Mathematical Sciences The Australian National UniversityCanberra ACT 0200Australia e-mail: [email protected]
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Abstract

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A holomorphic map ψ of the unit disk ito itself induces an operator Cψ on holomorphic functions by composition. We characterize bounded and compact composition operators Cψ on Qp spaces, which coincide with the BMOA for p = 1 and Bloch spaces for p > 1. We also give boundedness and compactness characterizations of Cψ from analytic function space X to Qp spaces, X = Dirichlet space D, Bloch space B or B0 = {f: f′ ∈ H∞}.

Keywords

composition operatorsQp spacesDirichlet spaceBMOABloch space

MSC classification

Secondary: 47B38: Operators on function spaces (general) 46E15: Banach spaces of continuous, differentiable or analytic functions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 70 , Issue 2 , April 2001 , pp. 161 - 188
Copyright
Copyright © Australian Mathematical Society 2001

References

[ArFiPe]Arazy, J., Fisher, S. and Peetre, J., ‘Möbius invariant function spaces’, J. Reine Angew. Math. 363 (1985), 110–145.Google Scholar
[AuLa]Aulaskari, R. and Lappan, P., ‘Criteria for an analytic function to be Bloch and a harmonic or meromorphic function to be normal’, in: Complex analysis and its applications, Pitman Research Notes in Math. 305 (Longman, 1994) pp. 136–146.Google Scholar
[AuStXi]Aulaskari, R., Stegenga, D. and Xiao, J., ‘Some subclasses of BMOA and their characterization in terms of Carleson measures’, Rocky Mountain J. Math. 26 (1996), 485–506.CrossRefGoogle Scholar
[AuXiZh]Aulaskari, R., Xiao, J. and Zhao, R., ‘On subspaces and subsets of BMOA and UBC’, Analysis 15 (1995), 101–121.CrossRefGoogle Scholar
[Ba]Baernstein, A., ‘Analytic functions of bounded mean oscillation’, in: Aspects of contemporary complex analysis (Academic Press, 1980) pp. 3–36.Google Scholar
[BoCiMa]Bourdon, P. S., Cima, J. A. and Matheson, A. L., ‘Compact composition operators on BMOA’, Trans. Amer. Math. Soc. 351 (1999), 2183–2196.CrossRefGoogle Scholar
[Co]Convey, J. B., Functions of one complex variable, 2nd edition (Springer, New York, 1978).Google Scholar
[EsXi]Essèn, M. and Xiao, J., ‘Q p spaces—A survey’, preprint.Google Scholar
[Ga]Garnett, J., Bounded analytic functions (Academic Press, 1981).Google Scholar
[Lo]Lou, Z., ‘Composition operators on Bloch type spaces’, preprint.Google Scholar
[MaMa]Madigan, K. and Matheson, A. L., ‘Compact composition operators on the Bloch spaces’, Trans. Amer. Math. Soc. 347 (1995), 2679–2687.Google Scholar
[Sh]Shapiro, J. H., ‘The essential norm of a composition operator’, Ann. of Math. (2) 125 (1987), 375–404.Google Scholar
[SmYa]Smith, W. and Yang, L., ‘Composition operators that improve integrability on weighted Bergman spaces’, Proc. Amer. Math. Soc. 126 (1998), 411–420.CrossRefGoogle Scholar
[SmZh]Smith, W. and Zhao, R., ‘Composition operators mapping into the Qp spaces’, Analysis 17 (1997), 239–263.CrossRefGoogle Scholar
[Tj]Tjani, M., Compact composition operators on some Mobius invariant Banach spaces (Ph.D. Thesis, Michigan State Univ., 1996).Google Scholar
[Xi]Xiao, J., ‘Carleson measure, atomic decomposition and free interpolation from Bloch space’, Ann. Acad. Sci. Fenn. Ser. A. I. Math. 19 (1994), 35–46.Google Scholar
[Zh]Zhu, K., Operator theory on function spaces (Marcel Dekker, New York, 1990).Google Scholar