On Bilinear Narrow Operators - Marat Pliev (MDPI AG, 2021)
In this article, we introduce a new class of operators on the Cartesian product of vector lattices. We say that a bilinear operator <semantics>T:E×F→W</semantics> 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 <semantics>Tx</semantics> and <semantics>Ty</semantics> are narrow for all <semantics>x∈E,y∈F</semantics>. 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 <semantics>E×F</semantics> 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 <semantics>E×F</semantics> of vector lattices E and F with the principal projection property to an order continuous Banach lattice G is narrow if and only if <semantics>|T|</semantics> is.