Composition operators from the space of Cauchy transforms to Bloch and the little Bloch-type spaces on the unit disk

Abstract

We completely characterize the boundedness and compactness of composition operators from the space of Cauchy transforms on the unit disk D, into the Bloch-type space Bν as well as the little Bloch-type space Bν,0, consisting respectively of all holomorphic functions f on D such that supz∈Dν(z)|f′(z)|<∞, that is, of all holomorphic functions f on D such that lim|z|→1ν (z)|f′(z)|=0, for some weight function ν. As a byproduct of our results, norm of the operator is calculated when Bν is replaced by Bν/C.