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

Abstract

The analytic self-map of the unit disk D, φ is said to induce a composition operator Cφ from the Banach space X to the Banach space Y if Cφ(f)=f∘φ∈Y for all f∈X. For z∈D and α>0, the families of weighted Cauchy transforms Fα are defined by f(z)=∫TKxα(z)dμ(x), where μ(x) is complex Borel measure, x belongs to the unit circle T, and the kernel Kx(z)=(1−x¯z)−1. In this paper, we will explore the relationship between the compactness of the composition operator Cφ acting on Fα and the complex Borel measures μ(x).