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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.

MSC:

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

Citations:

Zbl 0907.47032
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References:

[1] Abramovich, Y.A.; Aliprantis, C.D.; Burkinshaw, O., Invariant subspaces for positive operators, J. funct. analysis, 124, 95-111, (1994) · Zbl 0819.47049
[2] Abramovich, Y.A., C.D. Aliprantis and O. Burkinshaw — The invariant subspace problem: some recent advances. Rend. Istit. Mat. Univ. Trieste (forthcoming).
[3] Abramovich, Y.A.; Aliprantis, C.D.; Burkinshaw, O., Multiplication and compact-friendly operators, Positivity, 1, 171-180, (1997) · Zbl 0907.47032
[4] Aliprantis, C.D.; Burkinshaw, O., Positive operators, (1985), Academic Press New York & London · Zbl 0567.47037
[5] Lomonosov, V.I., Invariant subspaces of the family of operators that commute with a completely continuous operator, Funktsional. anal. i prilozhen, 7, No. 3, 55-56, (1973), (Russian)
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