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
)
→
C
0
(
Y
,
F
)
T : C_0(X, E) \to C_0(Y, F)
is compact (resp. completely continuous) if and only if
T
f
=
∑
n
δ
x
n
⊗
h
n
(
f
)
for all
f
∈
C
0
(
X
,
E
)
,
\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
→
B
(
E
,
F
)
h_n : Y \to B(E,F)
is continuous and vanishes at infinity in the uniform (resp. strong) operator topology, and
h
n
(
y
)
h_n(y)
is compact (resp.
h
n
h_n
is uniformly completely continuous).