Positivity

, Volume 22, Issue 1, pp 59–62

# On the “function” and “lattice” definitions of a narrow operator

Article

## 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).

### Keywords

Köthe Banach space Narrow operator Vector lattice

### Mathematics Subject Classification

Primary 46B20 Secondary 46B03 46B10

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