On the sum of a narrow and a compact operators

Our main technical tool is a principally new property of compact narrow operators which works for a domain space without an absolutely continuous norm. It is proved that for every Köthe F-space X and for every locally convex F-space Y the sum T1+T2 of a narrow operator T1:X→Y and a compact narrow operator T2:X→Y is a narrow operator. This gives a positive answers to questions asked by M. Popov and B. Randrianantoanina [6, Problems 5.6 and 11.63].