Abstract: In this paper we investigate narrow linear operators in the setting of Kothe-Bochner spaces of vector-valued measurable functions. We prove that for an atomless finite measure space \((A, \Sigma, \mu)\), a Kothe \(F\)-space \(E\) on \((A, \Sigma, \mu)\) with the absolutely continuous norm, a Banach space \(X\) and an \(F\)-space \(Y\) every bounded linear operator \(T \colon E(X) \to Y\) from a Kothe-Bochner space \(E(X)\) to \(Y\) is narrow, whenever the density of the range space \(Y\) is strictly less than the density of \(E(B)\) for every \(B \in \Sigma\) with \(\mu(B) > 0\). Our main result is based on methods and technique of the set-theoretic topology, the measure theory and the theory of \(F\)-spaces. In this connection, we mention the work of Popov, Semenov and Vatsek \cite{PSV}, where sufficient conditions on a rearrangement invariant space \(E\) were established for the orthogonal projection onto the closed linear span of the Rademacher system to be a hereditarily narrow operator. Moreover, in [1] the authors posed a number of open problems turned out to be important important for the further development of the theory of narrow operators.
For citation: Gutnova, A. K., Deunezev, R. R. and Pliev, M. A. On Narrow Operators on Kothe-Bochner Spaces, Vladikavkaz Math. J., 2025, vol. 28, no. 2, pp. 40-49. DOI 10.46698/g3050-4110-3527-j
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