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 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})