Abstract
We prove that the function and lattice definitions of a narrow operator defined on a Köthe Banach space E on a finite atomless measure space \((\Omega , \Sigma , \mu )\) are equivalent if and only if the set of all simple functions is dense in E. This answers Problem 10.3 from Popov and Randrianantoanina (Narrow operators on function spaces and vector lattices, De Gruyter studies in mathematics 45, De Gruyter, Berlin, 2013).
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Popov, M., Sobchuk, O. On the “function” and “lattice” definitions of a narrow operator. Positivity 22, 59–62 (2018). https://doi.org/10.1007/s11117-017-0497-6
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DOI: https://doi.org/10.1007/s11117-017-0497-6