Multipliers of classes of cauchy transforms

Abstract

The analytic map g on the unit disk D is said to induce a multiplication operator L from the Banach space X to the Banach space Y if L(f)=f·g∈Y for all f∈X. For z ∈ D and α>0 the families of weighted Cauchy transforms Fα are defined by ƒ(z) = ∫T Kx α (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 article we will explore the relationship between the compactness of the multiplication operator L acting on F 1 and the complex Borel measures μ(x). We also give an estimate for the essential norm of L