On Extension of Positive Multilinear Operators

Abstract

Using the linearization of positive multilinear operators by means of the Fremlin tensor product of vector lattices makes it possible to show that a multilinear operator from the Cartesian product of majorizing subspaces of vector lattices to Dedekind complete vector lattice admits extension to a positive multilinear operator on the Cartesian product of the ambient vector lattices. We establish that this is valid if the multilinear operator is defined on the Cartesian product of majorizing subspaces of separable Banach lattices and takes values in a topological vector lattice with the $\sigma$-interpolation property, provided that the Banach lattices have the property of subadditivity. The latter ensures that the algebraic tensor product of the majorizing subspaces is majorizing in the Fremlin tensor product of the Banach lattices. The open question is: whether or not the result is valid if the subadditivity property is omitted or weakened. The possibility of weakening the order completeness of the target lattice by some additional requirements on the initial vector lattices was firstly observed by Abramovich and Wickstead in proving a version of the Hahn–Banach–Kantorovich theorem.