Topological Properties of Weighted Composition Operators in Sequence Spaces

Source abstract

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