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)

ABSTRACT. Let X be an abstract set and .t; a lattice of subsets of X.

In this article, we introduce a new class of analytic functions of the unit disc D namely the Exponential Cauchy Transforms K e defined by f (z) = ∫ T exp [K (xz)] dµ(x) where K (z) = (1 − z) −1 is classical Cauchy kernel and µ(x) is a complex Borel measures and x belongs to the unit circle T. We use Laguerre polynomials to explore the coefficients of the Taylor expansions of the kernel and Peron's formula to study the asymptotic behavior of the Taylor coefficients. Finally we investigate relationships between our new class K e , the classical Cauchy space K and the Hardy spaces H p .

In this paper we prove a number of results on Cauchy transforms of generalized type given by Borel measures supported on the class of analytic functions mapping the unit disc into the unit disk. We also give a BMOA characterization using these families.2000 Mathematics subject classification: primary: 30E20; secondary: 30D99.

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