Points of narrowness and uniformly narrow operators

Abstract

It is known that the sum of every two narrow operators on $L_1$ is narrow, however the same is false for $L_p$ with $1 < p < \infty$. The present paper continues numerous investigations of the kind. Firstly, we study narrowness of a linear and orthogonally additive operators on Kothe function spaces and Riesz spaces at a fixed point. Theorem 1 asserts that, for every Kothe Banach space $E$ on a finite atomless measure space there exist continuous linear operators $S,T: E \to E$ which are narrow at some fixed point but the sum $S+T$ is not narrow at the same point. Secondly, we introduce and study uniformly narrow pairs of operators $S,T: E \to X$, that is, for every $e \in E$ and every $\varepsilon > 0$ there exists a decomposition $e = e' + e''$ to disjoint elements such that $\|S(e') - S(e'')\| < \varepsilon$ and $\|T(e') - T(e'')\| < \varepsilon$. The standard tool in the literature to prove the narrowness of the sum of two narrow operators $S+T$ is to show that the pair $S,T$ is uniformly narrow. We study the question of whether every pair of narrow operators with narrow sum is uniformly narrow. Having no counterexample, we prove several theorems showing that the answer is affirmative for some partial cases.