Let X be a compact metric space, and Lip(X) is the space of all bounded real-valued Lipschitz functions on X. A linear map T:Lip(X)->Lip(Y) is called disjointness preserving if fg=0 in Lip(X) implies TfTg=0 in Lip(Y). We prove that a biseparating linear bijection T(i.e. T and T^-1 are separating) is a weighted composition operator Tf=hf¡³£p, f is Lipschitz space from X onto R, £p is a homeomorphism from Y onto X, and h(y) is a Lipschitz function in Y.

Let X, Y be locally compact Hausdorff spaces. A linear operator T from C0(X,E) to C0(Y,F) is called disjointness preserving if coz(Tf)¡äcoz(Tg) = whenever coz(f)¡äcoz(g) = ∅. We discuss some cases on these compact disjointness preserving operators T and prove that if £f0 is a nonzero point of £m(T), then £f0 is an eigenvalue of T and we find a projection ∏: C0(X,E) ¡÷C0(X,E), such that for Y1 = ∏C0(X;E) and Y2 = (1-∏)C0(X;E), the operator T|Y1 -£f0 is a nilpotent and £f0-T|Y2 is invertible.