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

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) = ∫ T K x (z)dμ(x) where μ(x) is complex Borel measures, x belongs to the unit circle T and the kernel Kx(z) = (1 − xz) −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). A.M.S. (MOS) Subject Classification Codes. 30E20, 30D99.