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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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The Koebe semigroup and a class of averaging operators on $H^ p(\textbf {D})$
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by Aristomenis G. Siskakis PDF
Trans. Amer. Math. Soc. 339 (1993), 337-350 Request permission

Abstract:

We study on the Hardy space ${H^p}$ the operators ${T_F}$ given by \[ {T_F}(f)(z) = \frac {1} {z}\int _0^z {f(\zeta )\frac {1} {{F(\zeta )}}\;d\zeta } \] 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 ${\left \| {{T_F}} \right \|_p} \leq 2p\operatorname {Re} (1/F(0)) + |\operatorname {Im} (1/F(0))|$ 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
  • © Copyright 1993 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 339 (1993), 337-350
  • MSC: Primary 47B38; Secondary 30D55, 47D03
  • DOI: https://doi.org/10.1090/S0002-9947-1993-1147403-9
  • MathSciNet review: 1147403