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BELLWETHERS FOR BOUNDEDNESS OF COMPOSITION OPERATORS ON WEIGHTED BANACH SPACES OF ANALYTIC FUNCTIONS

Part of: Special classes of linear operators

Published online by Cambridge University Press:  01 June 2009

PAUL S. BOURDON
PAUL S. BOURDON*
Affiliation:
Department of Mathematics, Washington and Lee University, Lexington, VA 24450, USA (email: pbourdon@wlu.edu)
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Abstract

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Let 𝔻 be the open unit disc, let v:𝔻→(0,∞) be a typical weight, and let Hv∞ be the corresponding weighted Banach space consisting of analytic functions f on 𝔻 such that . We call Hv∞ a typical-growth space. For ϕ a holomorphic self-map of 𝔻, let Cφ denote the composition operator induced by ϕ. We say that Cφ is a bellwether for boundedness of composition operators on typical-growth spaces if for each typical weight v, Cφ acts boundedly on Hv∞ only if all composition operators act boundedly on Hv∞. We show that a sufficient condition for Cφ to be a bellwether for boundedness is that ϕ have an angular derivative of modulus less than 1 at a point on ∂𝔻. We raise the question of whether this angular-derivative condition is also necessary for Cφ to be a bellwether for boundedness.

Keywords

composition operatorweighted Bergman space of infinite orderboundednessangular derivativeLusky condition

MSC classification

Secondary: 47B33: Composition operators
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 86 , Issue 3 , June 2009 , pp. 305 - 314
Copyright
Copyright © Australian Mathematical Society 2009

References

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