Similarity to a contraction and hypercontractivity of composition operators

Jaoua, Nizar

doi:10.1090/S0002-9939-00-05843-3

Similarity to a contraction and hypercontractivity of composition operators
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Proc. Amer. Math. Soc. 129 (2001), 2085-2092 Request permission

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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