Extension of multilinear maps defined on subspaces
The lack of a Hahn-Banach theorem for multilinear forms has prompted a lot of research on the subject. In this interesting article, the standpoint considered is extendibility from subspaces, following the study initiated by \textit{M. Fernández-Unzueta} and \textit{A. Prieto} [Math. Proc. Camb. Philos. Soc. 148, No. 3, 505--518 (2010; Zbl 1197.46027)] for polynomials and holomorphic functions. Here, a product of Banach spaces \(E_1\times\cdots\times E_n\) is said to have the multilinear extension property if, whenever \(F_i\subset E_i\) are closed subspaces and \(T\in \mathcal L_n(F_1\times\cdots\times F_n)\) is a continuous \(n\)-linear mapping, then there exists a continuous \(n\)-linear map \(\widetilde T\in \mathcal L_n(E_1\times\cdots\times E_n)\) such that \(\widetilde T|_{F_1\times\cdots\times F_n}=T\). The main results proved in this article are the following: {\parindent=1 em\begin{itemize}\item[-] The multilinear extension property is inherited by subspaces and also by the product of some of the spaces. \item[-] The multilinear extension property is a local property: it is fully determined by the extendibility behavior of the \(n\)-linear forms defined on finite dimensional subspaces. \item[-] The multilinear extension property always forces control of the norms of the extensions. \item[-] The multilinear extension property imposes restrictions on the type and cotype of the spaces. In particular, if a Banach space \(E\) fails to have type \(p\) for some \(p<2\) or cotype \(q\) for some \(q>2\), then there exists \(n\) such that \(E\times\cdots\times E\) (\(n\) times) does not have the multilinear extension property. \end{itemize}} For the particular case \(n=2\), the property is naturally called bilinear extension property. By Maurey's extension theorem, if the Banach spaces \(E_1\) and \(E_2\) have type 2, then \(E_1\times E_2\) has the bilinear extension property. A ``weak converse'' of this statement is proved in this article: if \(E_1\times E_2\) has the bilinear extension property, then both \(E_1\) and \(E_2\) have weak type 2. Two open questions are left for further research. The first one asks if the bilinear extension property for \(E_1\times E_2\) forces \(E_1\) and \(E_2\) to have type 2. The other one has to do with the obvious fact that any product of Hilbert spaces has the multilinear extension property. So, the question is whether there exists a Banach space \(E\) not isomorphic to a Hilbert space such that, for every \(n\), \(E\times\cdots\times E\) (\(n\) times) has the multilinear extension property.
scientific article; zbMATH DE number 6105452
scientific article