Weighted composition operators from Bergman-type spaces into Bloch spaces

Let D be the open unit disk in the complex plane C. Denote by H(D) the class of all functions analytic on D. An analytic self-map φ : D → D induces the composition operator Cφ on H(D), defined by Cφ ( f ) = f (φ(z)) for f analytic on D. It is a well-known consequence of Littlewood’s subordination principle that the composition operator Cφ is bounded on the classical Hardy and Bergman spaces (see, for example [1]). Recall that a linear operator is said to be bounded if the image of a bounded set is a bounded set, while a linear operator is compact if it takes bounded sets to sets with compact closure. It is interesting to provide a function theoretic characterization of when φ induces a bounded or compact composition operator on various spaces. The book [1] contains plenty of information on this topic. Let u be a fixed analytic function on the open unit disk. Define a linear operator uCφ on the space of analytic functions on D, called a weighted composition operator, by uCφ f = u · ( f ◦φ), where f is an analytic function on D. We can regard this operator as a generalization of a multiplication operator and a composition operator. A positive continuous function φ on [0,1) is called normal, if there exist positive numbers s and t, 0 < s < t, such that