A characterization of compact-friendly multiplication operators

Answering in the affirmative a question posed in [3], 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.