Topological Properties of Weighted Composition Operators in Sequence Spaces

For fixed sequences u=(ui)i∈N,φ=(φi)i∈N , we consider the weighted composition operator Wu,φ with symbols u, φ defined by x=(xi)i∈N↦u(x∘φ)=(uixφi)i∈N . We characterize the continuity and the compactness of the operator Wu,φ when it acts on the weighted Banach spaces lp(v) , 1 ≤ p≤ ∞ , and c(v) , with v=(vi)i∈N a weight sequence on N . We extend these results to the case in which the operator Wu,φ acts on sequence (LF)-spaces of type lp(V) and on sequence (PLB)-spaces of type ap(V) , with p∈ [1 , ∞] ∪ { 0 } and V a system of weights on N . We also characterize other topological properties of Wu,φ acting on lp(V) and on ap(V) , such as boundedness, reflexivity and to being Montel. .