Extension of multilinear operators on Banach spaces

Abstract

This paper considers the problem of extending multilinear forms on a Banach space X to a larger space Y containing it as a closed subspace. For instance, if X is a subspace of Y and X0 ! Y 0 extends linear forms, then the induced Nicodemi operators extend multilinear forms. It is shown that an extension operator X0 ! Y 0 exists if and only if X is locally complemented in Y . Also, these extension operators preserve the symmetry if and only if X is regular. Finally, multlinear characterizations are obtained of some classical Banach space properties (Dunford-Pettis, etc.) related to weak compactness in terms of operators having Z-valued Aron-Berner extensions.