Compact composition operators on the Bloch space in polydiscs

Abstract

Let Uⁿ be the unit polydisc of ℂⁿ and φ=(φ₁, ⋯, φ _n ) a holomorphic self-map of Uⁿ. As the main result of the paper, it shows that the composition operator C is compact on the Bloch space β(Uⁿ) if and only if for every ε > 0, there exists a δ > 0, such that

$$\sum\limits_{k,1 = 1}^n {\left| {\frac{{\partial \phi _l }}{{\partial z_k }}(z)} \right|} \frac{{1 - |z_k |^2 }}{{1 - |\phi _l (z)|^2 }}< \varepsilon ,$$

whenever dist(φ(z), ∂U ⁿ)<δ.