On sums of narrow operators on Köthe function spaces
The paper deals with narrow operators in the sense of \textit{A. M. Plichko} and the second author [Diss. Math. 306, 85 p. (1990; Zbl 0715.46011)] defined on a space of functions on an atomless finite measure space \((\Omega, \Sigma, \mu)\) (see [the second author and \textit{B. Randrianantoanina}, Narrow operators on function spaces and vector lattices. Berlin: Walter de Gruyter (2013; Zbl 1258.47002)] for a detailed exposition of the subject). The authors solve two open problems. First, they answer in the negative the long standing question whether for every Banach space \(X\) a sum of two narrow operators from \(L_1[0, 1]\) to \(X\) must be narrow. Secondly, they demonstrate the existence of regular narrow operators \(T_1,T_2: L_\infty[0,1] \to L_\infty[0,1]\) with non-narrow sum \(T=T_1+T_2\). They also prove some interesting general results related to these problems.
scientific article; zbMATH DE number 6390625
scientific article