On two definitions of a narrow operator on Köthe–Bochner spaces

Abstract

We prove that for a Kothe–Banach space E with an order continuous norm over a finite atomless measure space \((\Omega ,\Sigma ,\mu )\) and for Banach spaces X, Y, the classes of narrow and weakly functionally narrow operators from a Kothe–Bochner space E(X) to a Banach space Y are coincident. We also obtain that in the general case, without the assumption of order continuity of the norm of E, the definitions of narrow and weakly functionally narrow operators from a Kothe–Bochner space E(X) to a Banach space Y are equivalent if and only if the set of all simple elements is dense in E(X).