Let X, Y be compact Hausdorff spaces
and E, F be Banach spaces.
A linear map  is separating if
T f, T g have disjoint cozeroes whenever
f, g have disjoint cozeroes. We prove that a
biseparating linear bijection T (that is, T
and  are separating) is a weighted composition
operator  . Here, h
is a function from Y
into the set of invertible linear operators from E
onto F, and
 is a homeomorphism from Y onto
X. We also show that T
is bounded if and only if h(y)
is a bounded operator from E
onto F for all y
in Y. In this case, h
is continuous with respect to the strong operator topology.
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