Let be the open unit disk in the complex plane and the class of all analytic functions on , and let be an analytic self-map, that is, . Then induces the composition operator on defined by for and . Let be a fixed analytic function on . The weighted composition operator is defined on by
Let denote the space of all such that
where the supremum is taken over all and . The little Zygmund space is defined by if and only if . An analytic function is said to belong to the Bloch space if . Finally, the closed subspace (the little Bloch space) of consists of the functions satisfying . Let and be the Banach spaces. For a given operator , the problem of investigating its boundedness and compactness is a natural and important problem of operator theory.
In this paper, the authors study the boundedness and compactness of weighted composition operators from Zygmund space into Bloch space and little Bloch space, based mainly on the classical methods of function theory.