Composition operators between growth spaces on circular and strictly convex domains in complex Banach spaces

Let ΩX be a bounded, circular and strictly convex domain in a complex Banach space X, and H(ΩX) be the space of all holomorphic functions from ΩX to C. The growth space A^ν (ΩX) consists of all f ∈ H(ΩX) such that |f(x)| ≤ Cν(rΩX (x)), x ∈ ΩX, for some constant C 0, whenever rΩX is the Minkowski functional on ΩX and ν : [0, 1) → (0, ∞) is a nondecreasing, continuous and unbounded function. For complex Banach spaces X and Y and a holomorphic map φ : ΩX → ΩY , put Cφ(f) = f ◦ φ, f ∈ H(ΩY ). We characterize those φ for which the composition operator Cφ : A ω (ΩY ) → A^ν (ΩX) is a bounded or compact operator.