Absolutely continuous measures and compact composition operator on spaces of Cauchy transforms

摘要

The analytic self-map of the unit diskD,φis said to induce a composition operatorCφfrom the Banach spaceXto the Banach spaceYifCφ(f)=f°φ∈Yfor allf∈X. Forz∈Dandα>0, the families of weighted Cauchy transformsFαare defined byf(z)=∫TKxα(z)dμ(x), whereμ(x)is complex Borel measure,xbelongs to the unit circleT, and the kernelKx(z)=(1?xˉz)?1. In this paper, we will explore the relationship between the compactness of the composition operatorCφacting onFαand the complex Borel measuresμ(x).