Properties of superposition operators acting between Bμ∗ and QK∗

In this paper we introduce natural metrics in the hyperbolic Bloch and QKQK-type spaces with respect to which these spaces are complete. Moreover, Lipschitz continuous, bounded and compact superposition operators SϕSϕ from the hyperbolic Bloch type space to the hyperbolic QKQK-type space are characterized by conditions depending only on the analytic symbol ϕϕ.