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

Abstract

We study on the Hardy space H p {H^p} the operators T F {T_F} given by \[ T F ( f ) ( z ) = 1 z ∫ 0 z f ( ζ ) 1 F ( ζ ) d ζ {T_F}(f)(z) = \frac {1} {z}\int _0^z {f(\zeta )\frac {1} {{F(\zeta )}}\;d\zeta } \] where F ( z ) F(z) is analytic on the unit disc D \mathbb {D} and has Re ⁡ F ( z ) ≥ 0 \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 ‖ T F ‖ p ≤ 2 p Re ⁡ ( 1 / F ( 0 ) ) + | Im ⁡ ( 1 / F ( 0 ) ) | {\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 {T_F} is compact on H p {H^p} if and only if the measure μ \mu in the Herglotz representation of 1 / F 1/F is continuous.