Weighted composition operators on spaces of functions with derivative in a Bergman space

Abstract

Following the idea of the paper by Contreras and Hernandez-Diaz, we first give the characterization of the boundedness of the weighted composition operator $$W_{\varphi ,\phi }$$ on $$B^p$$ . Then, we investigate the boundedness and the compactness of composition operators $$C_{\varphi}$$ on $$B^p$$ and study the spectrum of the multiplication operator $$M_{\phi}$$ on $$B^p$$ , as well. Finally, motivated by the paper by Cuckovic and Paudyal, we describe the relationships between the invariant subspaces of $$M_{z}$$ on $$B^p_{0}$$ and T on $$A^p$$ , where T is the sum of multiplication operator and Volterra operator. Moreover, we provide some Beurling-type invariant subspaces of $$M_{z}$$ on $$B^p$$ and $$B^p_0$$ , respectively.