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The domain space of an analytic composition operator

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

Published online by Cambridge University Press:  09 April 2009

Thomas Domenig ,
Hans Jarchow  and
Reinhard Riedl
Thomas Domenig
Affiliation:
Institut für Mathematik, Universität Zürich, Winterthurerstr. 190, CH 8057 Zürich, Switzerland e-mail: domenig@amath.unizh.ch, jarchow@math.unizh.ch, riedl@ifi.unizh.ch
Hans Jarchow
Affiliation:
Institut für Mathematik, Universität Zürich, Winterthurerstr. 190, CH 8057 Zürich, Switzerland e-mail: domenig@amath.unizh.ch, jarchow@math.unizh.ch, riedl@ifi.unizh.ch
Reinhard Riedl
Affiliation:
Institut für Mathematik, Universität Zürich, Winterthurerstr. 190, CH 8057 Zürich, Switzerland e-mail: domenig@amath.unizh.ch, jarchow@math.unizh.ch, riedl@ifi.unizh.ch
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Abstract

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In this paper we show that, for analytic composition operators between weighted Bergman spaces (including Hardy spaces) and as far as boundedness, compactness, order boundedness and certain summing properties of the adjoint are concerned, it is possible to modify domain spaces in a systematic fashion: there is a space of analytic functions which embeds continuously into each of the spaces under consideration and on which the above properties of the operator are decided.

A remarkable consequence is that, in the setting of composition operators between weighted Bergman spaces, the properties in question can be identified as properties of the operator as a map between appropriately chosen Hilbert spaces.

Keywords

Composition operatorsweighted Bergman spacesHardy spacesboundednesscompactnessabsolutely summing operators

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 66 , Issue 1 , February 1999 , pp. 56 - 65
Copyright
Copyright © Australian Mathematical Society 1999

References

[1]MacCluer, C. Cowen-B., Composition operators on spaces of analytic functions (CRC Press, Boca Raton, 1995).Google Scholar
[2]Diestel, J., Sequences and series in Banach spaces (Springer, New York, 1984).CrossRefGoogle Scholar
[3]Diestel, J., Jarchow, H. and Tonge, A., Absolutely summing operators (Cambridge University Press, Cambridge, 1995).CrossRefGoogle Scholar
[4]Domenig, T., Composition operators on weighted Bergman spaces and Hardy spaces (Dissertation, Universität Zürich, 1997).Google Scholar
[5]Domenig, T., ‘Order bounded and p-summing composition operators’, Contemp. Math. 213 (1998), 27–41.CrossRefGoogle Scholar
[6]Duren, P., Theory of Hp spaces (Academic Press, New York, 1970).Google Scholar
[7]Hunziker, H. and Jarchow, H., ‘Composition operators which improve integrability’, Math. Nachr. 122 (1991), 83–99.CrossRefGoogle Scholar
[8]Jarchow, H., ‘Absolutely summing composition operators’, in: Proc. Conf. Funct. Anal. (Marcel Dekker, 1993) pp. 193–203.Google Scholar
[9]Jarchow, H., ‘Compactness properties of composition operators’, submitted.Google Scholar
[10]Jarchow, H. and Riedl, R., ‘Factorization of composition operators through Bloch type spaces’, Illinois J. Math. 39 (1995), 431–440.CrossRefGoogle Scholar
[11]Luecking, D. H., ‘Representation and duality in weighted spaces of analytic functions’, Indiana Univ. Math. J. 34 (1985), 319–336.CrossRefGoogle Scholar
[12]Pietsch, A., Operator ideals (North-Holland, Amsterdam, 1980).Google Scholar
[13]Riedl, R., Composition operators and geometric properties of analytic functions (Dissertation, Universität Zürich, 1994).Google Scholar
[14]Shapiro, J. H., ‘Mackey topologies, reproducing kernels, and diagonal maps on the Hardy and Bergman spaces’, Duke Math. J. 43 (1976), 187–202.CrossRefGoogle Scholar
[15]Shapiro, J. H., Composition operators and classical function theory (Springer, New York, 1993).CrossRefGoogle Scholar
[16]Shapiro, J. H. and Taylor, P. D., ‘Compact, nuclear, and Hilbert-Schmidt composition operators on H2’, Indiana Univ. Math. J. 23 (1973), 471–496.CrossRefGoogle Scholar
[17]Sledd, W. T., ‘Some results about spaces of analytic functions introduced by Hardy and Littlewood’, Michigan Math. J. 31 (1984), 307–319.Google Scholar
[18]Smith, W., ‘Composition operators between Bergman and Hardy spaces’, Trans. Amer Math. Soc. 348 (1996), 2331–2348.CrossRefGoogle Scholar
[19]Zhu, K., Operator theory in function spaces (Marcel Dekker, New York, 1990).Google Scholar