Sums of generalized weighted composition operators from weighted Bergman spaces to weighted Banach spaces

Abstract

The present research provides both necessary and sufficient conditions for the sum operator \mathcal{S}_{\mu,\eta}^k to exhibit boundedness and compactness when mapping from the weighted Bergman spaces \mathcal{A}_v^p to the weighted Banach spaces H_w^\infty(H_w^0). This unification encompasses the product of multiplication, differentiation, and composition operators. Furthermore, we provide an example to demonstrate that the boundedness of the operator \mathcal{S}_{\mu,\eta}^k:\mathcal{A}_v^p\longrightarrow H_w^\infty does not necessarily imply the boundedness of the operator \mathcal{S}_{\mu,\eta}^k:\mathcal{A}_v^p\longrightarrow H_w^0. Also, we present an example of a bounded operator \mathcal{S}_{\mu,\eta}^k:H_v^\infty\longrightarrow H_w^\infty, while the operator \mathcal{S}_{\mu,\eta}^k:\mathcal{A}_v^p\longrightarrow H_w^\infty is not bounded.