Weighted composition operators between weak spaces of vector-valued analytic functions

Abstract

Abstract We consider weighted composition operators W ψ , φ : f ↦ ψ ⁢ ( f ∘ φ ) ${W_{\psi,\varphi}\colon f\mapsto\psi(f\circ\varphi)}$ on spaces of analytic functions on the unit disc, which take values in some complex Banach space. We provide necessary and sufficient conditions for the boundedness and (weak) compactness of W ψ , φ ${W_{\psi,\varphi}}$ on general function spaces, and in particular on weak vector-valued spaces. As an application, we characterize the weak compactness of W ψ , φ ${W_{\psi,\varphi}}$ between two different vector-valued Bloch-type spaces. This characterization appears to be new also in the scalar-valued case.