Extension of multilinear maps defined on subspaces
type: Item
instance of: scholarly article
title: Extension of multilinear maps defined on subspaces
zbMATH Open document ID: 1270.46039
published in: Israel Journal of Mathematics
publication date: 2012-11-13T00:00:00Z
review text: 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 t…
reviewed by: Verónica Dimant
Mathematics Subject Classification ID: 46G25
Mathematics Subject Classification ID: 46A22
Mathematics Subject Classification ID: 47A07
Mathematics Subject Classification ID: 46B07
zbMATH DE Number: 6105452
zbMATH Keywords: multilinear mappings
zbMATH Keywords: extendibility from subspaces
zbMATH Keywords: multilinear extension property
zbMATH Keywords: bilinear extension property
zbMATH Keywords: weak type 2
author: Maite Fernández-Unzueta
MaRDI profile type: Publication
full work available at URL: s11856-011-0151-0
OpenAlex ID: W1994966460
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DOI: 10.1007/S11856-011-0151-0
description: scientific article; zbMATH DE number 6105452