On the sum of a narrow and a compact operators

Volodymyr Mykhaylyuk

paper

On the sum of a narrow and a compact operators

arXiv:1512.07838

Abstract

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 $T_{1} + T_{2}$ 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})