Topological Properties of Weighted Composition Operators in Sequence Spaces

Abstract

For fixed sequences u = (u_i)_{iin {{mathbb {N}}}}, varphi =(varphi _i)_{iin {{mathbb {N}}}}, we consider the weighted composition operator W_{u,varphi } with symbols u, varphi defined by x=(x_i)_{iin {{mathbb {N}}}}mapsto u(xcirc varphi )= (u_ix_{varphi _i})_{iin {{mathbb {N}}}}. We characterize the continuity and the compactness of the operator W_{u,varphi } when it acts on the weighted Banach spaces l^p(v), 1le ple infty , and c_0(v), with v=(v_i)_{iin {{mathbb {N}}}} a weight sequence on {{mathbb {N}}}. We extend these results to the case in which the operator W_{u,varphi } acts on sequence (LF)-spaces of type l_p(mathcal {V}) and on sequence (PLB)-spaces of type a_p(mathcal {V}), with pin [1,infty ] cup {0} and mathcal {V} a system of weights on {{mathbb {N}}}. We also characterize other topological properties of W_{u,varphi } acting on l_p(mathcal {V}) and on a_p(mathcal {V}), such as boundedness, reflexivity and to being Montel.