$\mathbb{N}$-compactness and weighted composition maps

In this paper we study some conditions on (not necessarily continuous) linear maps $T$ between spaces of real- or complex-valued continuous functions $C (X)$ and $C (Y)$ which allow us to describe them as weighted composition maps. This description depends strongly on the topology in $X$; namely, it can be given when $X$ is $\mathbb{N}$-compact, but cannot in general if some kind of connectedness on $X$ is assumed. Finally we also give an infimum-preserving version of the Banach-Stone theorem. The results are also proved for spaces of bounded continuous functions when $\mathbb{K}$ is a field endowed with a nonarchimedean valuation and it is not locally compact.