Composition Operators on Spaces of Fractional Cauchy Transforms

For ?? > 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 ?? be an analytic self-map of the unit disc D. The map ?? 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 ?? are given, sufficient to imply that C _?? is bounded on ${mathcal{F}_{alpha}}$ , in the case 0 < ?? < 1. Several of the conditions involve ???? and the theory of multipliers of the space ${mathcal{F}_{alpha}}$ . Relations are found between the behavior of C _?? and the membership of ?? in the Dirichlet spaces. Conditions given in terms of the generalized Nevanlinna counting function are shown to imply that ?? induces a bounded composition operator on ${mathcal{F}_{alpha}}$ , in the case 1/2 ?? ?? < 1. For such ??, examples are constructed such that ${| Phi |_{infty} = 1}$ and ${C_{Phi}: mathcal{F}_{alpha} rightarrow mathcal{F}_{alpha}}$ is bounded.