Characterizations of Disjointness preserving operators on vector-valued function spaces

Abstract:

We characterize compact and completely continuous disjointness preserving linear operators on vector-valued continuous functions as follows: a disjointness preserving operator $T : C_0(X, E) \to C_0(Y, F)$ is compact (resp. completely continuous) if and only if \begin{align*} Tf = \sum _n \delta _{x_n} \otimes h_n (f) \quad \text {for all } f \in C_0(X,E), \end{align*} where $h_n : Y \to B(E,F)$ is continuous and vanishes at infinity in the uniform (resp. strong) operator topology, and $h_n(y)$ is compact (resp. $h_n$ is uniformly completely continuous).