Range preserving maps between the spaces of continuous functions with values in a locally convex space

Let $C(X,E)$ be the linear space of all continuous functions on a compact Hausdorff space $X$ with values in a locally convex space $E$. We characterize maps $T:C(X,E)\to C(Y,E)$ which satisfy $\mathrm{R}\mathrm{a}\mathrm{n}(TF-TG)\subset \mathrm{R}\mathrm{a}\mathrm{n}(F-G)$ for all $F,G\in C(X,E)$.