Communications of the
Korean Mathematical Society CKMS

Let $A$ be a normed space, $\mathcal B(A)$ the algebra of all bounded operators on $A$, and $V$ a family of strongly upper semicontinuous functions from a Hausdorff completely regular space $X$ into $\mathcal B(A)$. In this paper, we investigate some properties of the weighted spaces $CV(X, A)$ of all $A$-valued continuous functions $f$ on $X$ such that the mapping $x \mapsto v(x)(f(x))$ is bounded on $X$, for every $v \in V$, endowed with the topology generated by the seminorms $\|f\|_v = \sup\{\|v(x)(f(x))\| ,\ x \in X\}$. Our main purpose is to characterize continuous, bounded, and locally equicontinuous weighted composition operators between such spaces.

Keywords: generalized Nachbin family, generalized weighted spaces of vector-valued continuous functions, weighted composition operators, multiplication operators

MSC numbers: Primary 46E10, 47B38; Secondary 47A56, 46E40