Topological Properties of Weighted Composition Operators in Sequence Spaces

Abstract

AbstractFor fixed sequences$$u = (u_i)_{i\in {{\mathbb {N}}}}, \varphi =(\varphi _i)_{i\in {{\mathbb {N}}}}$$u=(ui)i∈N,φ=(φi)i∈N, we consider the weighted composition operator$$W_{u,\varphi }$$Wu,φwith symbols$$u, \varphi $$u,φdefined by$$x=(x_i)_{i\in {{\mathbb {N}}}}\mapsto u(x\circ \varphi )= (u_ix_{\varphi _i})_{i\in {{\mathbb {N}}}}$$x=(xi)i∈N↦u(x∘φ)=(uixφi)i∈N. We characterize the continuity and the compactness of the operator$$W_{u,\varphi }$$Wu,φwhen it acts on the weighted Banach spaces$$l^p(v)$$lp(v),$$1\le p\le \infty $$1≤p≤∞, and$$c_0(v)$$c0(v), with$$v=(v_i)_{i\in {{\mathbb {N}}}}$$v=(vi)i∈Na weight sequence on$${{\mathbb {N}}}$$N. We extend these results to the case in which the operator$$W_{u,\varphi }$$Wu,φacts on sequence (LF)-spaces of type$$l_p(\mathcal {V})$$lp(V)and on sequence (PLB)-spaces of type$$a_p(\mathcal {V})$$ap(V), with$$p\in [1,\infty ] \cup \{0\}$$p∈[1,∞]∪{0}and$$\mathcal {V}$$Va system of weights on$${{\mathbb {N}}}$$N. We also characterize other topological properties of$$W_{u,\varphi }$$Wu,φacting on$$l_p(\mathcal {V})$$lp(V)and on$$a_p(\mathcal {V})$$ap(V), such as boundedness, reflexivity and to being Montel.