Abstract
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 Tx and Ty 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.
Highlights
Linear narrow operators on function spaces can be considered as a generalization of compact operators
The vector space of all bounded bilinear operators from the Cartesian product E × F of normed spaces E and F to a Banach space W we denote by B(E, F; W)
We note that a bilinear operator T : E × F → W does not need to be linear as an operator defined on a vector lattice E × F
Summary
Linear narrow operators on function spaces can be considered as a generalization of compact operators. The vector space of all bounded bilinear operators from the Cartesian product E × F of normed spaces E and F to a Banach space W we denote by B(E, F; W). We note that a bilinear operator T : E × F → W does not need to be linear as an operator defined on a vector lattice E × F. We observe that the notion of a narrow operator on a Köthe–Banach space can be extended to the setting of operators defined on vector lattices. A linear operator S : E → X is called narrow, if, for every x ∈ E and ε > 0, there exist mutually complemented fragments x1 and x2 of x, such that S(x1 − x2) < ε. Let E and F be vector lattices, X be a normed space and T : E × F → X be a bilinear operator. We note that the problem of coincidence of classes of function narrow and function weakly operators still remains open, even for the linear case (see [4])
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