On the sum of a narrow and a compact operators

TL;DR

This paper introduces a new property of compact narrow operators that allows proving the sum of a narrow and a compact narrow operator remains narrow, answering open questions in the theory of narrow operators.

Contribution

It establishes that the sum of a narrow operator and a compact narrow operator is narrow in certain F-spaces, using a novel property applicable without an absolutely continuous norm.

Findings

Sum of narrow and compact narrow operators is narrow in K"{o}the F-spaces.

Provides a positive answer to open problems by Popov and Randrianantoanina.

Introduces a new property of compact narrow operators applicable to a broader class of spaces.

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 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})