Composition Operators between Some Classical Spaces of Analytic Functions

An analytic self-map φ : D → D of the open unit disk D in the complex plane induces the composition operator Cφ on H(D), the space of holomorphic functions on D, defined by Cφ(f) = f ◦φ. A basic goal in the study of composition operators is to relate function theoretic properties of φ to operator theoretic properties of Cφ. Here we review some results that characterize when φ induces a bounded or compact composition operator between various classical Banach spaces of holomorphic functions on D. Recall that a linear operator T : X → Y is said to be bounded if the image of a bounded set in X is a bounded subset of Y , while T is compact if it takes bounded sets to sets with compact closure. Emphasis will be given to motivating the theorems and explaining relationships between them. Some proofs will be given, but most will be skipped, or at most indicated. Details of the proofs can be found in the literature cited.