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The domain space of an analytic composition operator

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

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

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