In the paper we study the operators on the weighted Bergman spaces on the unit disk \({\mathbb {D}}\) , denoted by \(A^{p}_{\lambda ,w}({\mathbb {D}})\) , that are associated with a class of generalized analytic functions, named the \(\lambda \) -analytic functions, and with a class of radial weight functions w. The main result is to give a characterization of the boundedness of a linear operator mapping \(A^{p}_{\lambda ,w}({\mathbb {D}})\) for \(2\lambda /(2\lambda +1)\le p\le 1\) into a Banach space by means of the behaviour near the boundary of a single vector-valued \(\lambda \) -analytic function related to the operator. As applications, we obtain a necessary and sufficient condition of sequence multipliers on the space \(A^{p}_{\lambda ,w}({\mathbb {D}})\) , and also give a sufficient condition of Carleson type for boundedness of multiplication operators on \(A^{p}_{\lambda ,w}({\mathbb {D}})\) associated with power weights.
