Composition Operators on Spaces of Fractional Cauchy Transforms

Abstract

For α > 0, the Banach space ${\mathcal{F}_{\alpha}}$ is defined as the collection of functions f which can be represented as integral transforms of an appropriate kernel against a Borel measure defined on the unit circle T. Let Φ be an analytic self-map of the unit disc D. The map Φ induces a composition operator on ${\mathcal{F}_{\alpha}}$ if ${C_{\Phi}(f) = f \circ \Phi \in \mathcal{F}_{\alpha}}$ for any function ${f \in \mathcal{F}_{\alpha}}$. Various conditions on Φ are given, sufficient to imply that C _Φ is bounded on ${\mathcal{F}_{\alpha}}$, in the case 0 < α < 1. Several of the conditions involve Φ′ and the theory of multipliers of the space ${\mathcal{F}_{\alpha}}$. Relations are found between the behavior of C _Φ and the membership of Φ in the Dirichlet spaces. Conditions given in terms of the generalized Nevanlinna counting function are shown to imply that Φ induces a bounded composition operator on ${\mathcal{F}_{\alpha}}$, in the case 1/2 ≤ α < 1. For such α, examples are constructed such that ${\| \Phi \|_{\infty} = 1}$ and ${C_{\Phi}: \mathcal{F}_{\alpha} \rightarrow \mathcal{F}_{\alpha}}$ is bounded.