A lattice approach to narrow operators (Q839537)

The authors extend the notion of a narrow operator to the general case of vector lattice. They prove that the set \(N_{r} (E,F)\) of all narrow regular operators forms a band in the vector lattice \(L_{r} (E,F)\) of all regular operators from a non-atomic order continuous Banach lattice \(E\) to an order continuous Banach lattice \(F\). In view of this fact, it is pointed out that the set \(N_{r} (L_{\infty})\) of all narrow regular operators in the space \(L_{\infty}\) is not a band in the vector lattice \(N_{r} (L_{\infty})\) of all regular linear operators in \(L_{\infty}\). The proposed approach allows to obtain various generalizations. In particular, an extension of the Kalton-Rosenthal representation theorem to the general case of vector lattices are given.