Composition Operators on Large Fractional Cauchy Transform Spaces

Abstract

For α >0 and z in the unit disk D the spaces of fractional Cauchy transforms F _α are known as the family of all functions f(z) such that f(z)= $\int_{T}[K(\overline{x}z)]^{\alpha}d\mu(x)$ where K(z)=(1-z)^-1 is the Cauchy kernel, T is the unit circle and μ ∈ $\mathcal{M}$ the set of complex Borel measure on T. The Banach space F _α may be written as F _α =(F _α)_a ⊕ (F _α)_s, where (F _α)_a is isomorphic to a closed subspace of $\mathcal{M}_a$ the subset of absolutely continuous measures of $\mathcal{M}$, and (F _α)_s is isomorphic to $\mathcal{M}_s$ the subspace of $\mathcal{M}$ of singular measures. In this article we show that for α ≥1, the composition operator C _φ is compact on K _α C _φ $C_\varphi[K^{\alpha}(\overline{x}z)]\subset(F_{\alpha})_a$ and in doing so, extend a result due to [1] who showed that C _φ is compact on F ₁ if and only if C _φ (F ₁) ⊂ (F ₁)_a.

Mathematics Subject Classification (2010). Primary: 30E20; Secondary: 30D99.