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