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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The Koebe semigroup and a class of averaging operators on $ H\sp p({\bf D})$


Author: Aristomenis G. Siskakis
Journal: Trans. Amer. Math. Soc. 339 (1993), 337-350
MSC: Primary 47B38; Secondary 30D55, 47D03
MathSciNet review: 1147403
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Abstract: We study on the Hardy space $ {H^p}$ the operators $ {T_F}$ given by

$\displaystyle {T_F}(f)(z) = \frac{1} {z}\int_0^z {f(\zeta )\frac{1} {{F(\zeta )}}\;d\zeta } $

where $ F(z)$ is analytic on the unit disc $ \mathbb{D}$ and has $ \operatorname{Re} F(z) \geq 0$. Each such operator is closely related to a strongly continuous semigroup of weighted composition operators. By studying first an extremal such semigroup (the Koebe semigroup) we are able to obtain the upper bound $ {\left\Vert {{T_F}} \right\Vert _p} \leq 2p\operatorname{Re} (1/F(0)) + \vert\operatorname{Im} (1/F(0))\vert$ for the norm. We also show that $ {T_F}$ is compact on $ {H^p}$ if and only if the measure $ \mu $ in the Herglotz representation of $ 1/F$ is continuous.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1993-1147403-9
PII: S 0002-9947(1993)1147403-9
Article copyright: © Copyright 1993 American Mathematical Society