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Similarity to a contraction and hypercontractivity of composition operators

Author(s): Nizar Jaoua
Journal: Proc. Amer. Math. Soc. 129 (2001), 2085-2092.
MSC (1991): Primary 47B38, 47B65
Posted: December 28, 2000
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Abstract: On the Hardy spaces $H^p$ with $1 \leq p<\infty$, we consider the composition operators induced by analytic self-maps of the open unit disc $D$. First, we characterize those which are similar to contractions. Then, we give some necessary and sufficient conditions for them to be hypercontractive. Finally, we prove that, among those ones, only the zero-symbol composition operator sends $H^p$ into $H^{\infty}$ with a norm less than or equal to $1$.

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Additional Information:

Nizar Jaoua
Affiliation: Department of Mathematics, Université Lille I, Cité Scientifique, F-59655 Villeneuve d'Ascq, France
Email: jaoua@agat.univ-lille1.fr

DOI: 10.1090/S0002-9939-00-05843-3
PII: S 0002-9939(00)05843-3
Keywords: Composition operators, Hardy spaces, contraction, hypercontractive (or $\beta$-contractive) operator
Received by editor(s): March 10, 1999
Received by editor(s) in revised form: November 17, 1999
Posted: December 28, 2000
Communicated by: David R. Larson
Copyright of article: Copyright 2000, American Mathematical Society

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