On two definitions of a narrow operator on Köthe-Bochner spaces
Let \((\Omega,\Sigma,\mu)\) be a finite atomless measure space, \(E\) be a Köthe-Banach space over \((\Omega,\Sigma,\mu)\), \(X\) be a Banach space and \(E(X)\) be the corresponding Köthe-Bochner space. Given a Banach space \(Y\), a continuous linear operator \(T : E(X) \to Y\) is called weakly functionally narrow if, for every \(A \in \Sigma\), \(x \in X\) and \(\varepsilon > 0\), there exists a decomposition of \(A\) into the disjoint union of \(B, C \in \Sigma\) in such a way that \(\|T(x \mathbf 1_B - x \mathbf 1_D)\| < \varepsilon\). The operator \(T\) is called narrow if, for every \(f \in E(X)\) and \(\varepsilon > 0\), there exist two mutually complemented fragments \(g\) and \(h\) of \(f\) such that \(\|T(g - h)\| < \varepsilon\). These definitions, given by the authors, generalize those ones that were previously known for \(X = \mathbb R\). It is demonstrated that, if the norm of \(E\) is order continuous, then the classes of narrow and weakly functionally narrow operators from \(E(X)\) to \(Y\) are the same. In the general case, without the assumption of order continuity, for given \(E(X)\) and \(Y\), the classes of narrow and weakly functionally narrow operators from \(E(X)\) to \(Y\) coincide if and only if the set of all simple elements is dense in \(E(X)\). The reader should take caution with the name ``narrow operator'', because in different papers it sometimes has similar but different meanings; for example, the paper [\textit{K. Boyko} et al., Zh. Mat. Fiz. Anal. Geom. 2, No. 4, 358--371 (2006; Zbl 1147.46006)] that addresses an analogous problem, works with a different non-equivalent definition.
scientific article; zbMATH DE number 6916266
scientific article