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

Abstract

Following the idea of the paper by Contreras and Hernández-Díaz, 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 Čučković 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.