A characterization of compact-friendly multiplication operators. (English) Zbl 1043.47504

Summary: Answering in the affirmative a question posed in [Y. A. Abramovich; C. D. Aliprantis and O. Burkinshaw, Positivity 1, 171–180 (1997; Zbl 0907.47032)], 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.


47B38 Linear operators on function spaces (general)
47B60 Linear operators on ordered spaces


Zbl 0907.47032
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