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

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-phi,W-phi on B-p. Then, we investigate the boundedness and the compactness of composition operators C-phi 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-0(p) 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-0(p), respectively.