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

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.