  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 K\"{o}the $F$-space $X$ and for every locally
convex $F$-space $Y$ the sum $T_1+T_2$ of a narrow operator $T_1:X\to Y$ and a
compact narrow operator $T_2:X\to Y$ is a narrow operator. This gives a
positive answers to questions asked by M.~Popov and B.~Randrianantoanina
(\cite[Problem 5.6 and Problem 11.63]{PR})
