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

Published online by Cambridge University Press:  01 June 2009

PAUL S. BOURDON*
Affiliation:
Department of Mathematics, Washington and Lee University, Lexington, VA 24450, USA (email: [email protected])
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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.

MSC classification

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 2009

References

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