On characterizations of weighted composition opeartors on non-locally convex weighted spaces of continuous functions

For a system $V$ of weights on a completely regular Hausdorff space $X$ and a Hausdorff topological vector space $E$, let $ CV_b(X,E)$ and $ CV_0(X,E)$ respectively denote the weighted spaces of continuouse $E$-valued functions $f$ on $X$ for which $ (vf)(X)$ is bounded in $E$ and $vf$ vanishes at infinity on $X$ all $ v\in V$. On $CV_b(X,E)(CV_0(X,E))$, consider the weighted topology, which is Hausdorff, linear and has a base of neighbourhoods of 0 consising of all sets of the form: $ N(v,G)=\{f:(vf)(X)\subseteq G\}$, where $v\in V$ and $G$ is a neighbourhood of 0 in $E$. In this paper, we characterize weighted composition operators on weighted spaces for the case when $V$ consists of those weights which are bounded and vanishing at infinity on $X$. These results, in turn, improve and extend some of the recent works of Singh and Singh [10, 12] and Manhas [6] to a non-locally convex setting as well as that of Singh and Manhas [14] and Khan and Thaheem [4] to a larger class of operators.