  Answering in the affirmative a question posed in [Y.A.Abramovich,
C.D.Aliprantis and O.Burkinshaw, Multiplication and compact-friendly operators,
Positivity 1 (1997), 171--180], we prove that a positive multiplication
operator on any $L_p$-space (resp. on a $C(\Omega)$-space) is compact-friendly
if and only if the multiplier is constant on a set of positive measure (resp.
on a non-empty open set).
  In the process of establishing this result, we also prove that any
multiplication operator has a family of hyperinvariant bands -- a fact that
does not seem to have appeared in the literature before. This provides useful
information about the commutant of a multiplication operator.
