On Bilinear Narrow Operators

In this article, we introduce a new class of operators on the Cartesian product of vector lattices. We say that a bilinear operator T : E × F → W defined on the Cartesian product of vector lattices E and F and taking values in a vector lattice W is narrow if the partial operators T x and T y are narrow for all x ∈ E , y ∈ F . We prove that, for order-continuous Köthe–Banach spaces E and F and a Banach space X , the classes of narrow and weakly function narrow bilinear operators from E × F to X are coincident. Then, we prove that every order-to-norm continuous C -compact bilinear regular operator T is narrow. Finally, we show that a regular bilinear operator T from the Cartesian product E × F of vector lattices E and F with the principal projection property to an order continuous Banach lattice G is narrow if and only if | T | is.