A characterization of compact-friendly multiplication operators

AbstractAnswering in the affirmative a question posed in [3], we prove that a positive multiplication operator on any Lp-space (resp. on a C(Ω)-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