A short proof of the result on actions that characterize \(\ell^n_\infty\)
type: Item
instance of: scholarly article
title: A short proof of the result on actions that characterize \(\ell^n_\infty\)
zbMATH Open document ID: 0942.46003
DOI: 10.1016/S0024-3795(99)00078-6
published in: Linear Algebra and its Applications
publication date: 1999-11-29T00:00:00Z
review text: The author gives a proof of a theorem due to \textit{B. Chalmers} and \textit{B. Shekhtman} [Linear Algebra Appl. 270, 155-169 (1998; Zbl 0898.46011)] respectively: Let \(T\) be an operator on an \(n\)-dimensional linear space \(L\) which is not a multiple of the identity. Then \(L\) can be endowed with a norm satisfying the following conditions: (1) The obtained \(n\)-dimensional normed space \(V\) is not isometric to \(\ell^n_\infty\); (2) For every isometric embedding of the space \(V\) into a Banach space \(X\) there exists a norm preserving extension \(\widetilde{T}\) of \(T\) onto \(X\). If the identity operator on an \(n\)-dimensional Banach space \(V\) can be extended to any Banach space with the same norm, then \(V\) is isometric to \(\ell^n_\infty\), and, the identity is the only such operator.
reviewed by: Silviu Crăciunaş
Mathematics Subject Classification ID: 46B07
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 52A21
Mathematics Subject Classification ID: 15A60
Mathematics Subject Classification ID: 46B03
zbMATH DE Number: 1371399
zbMATH Keywords: norm preserving extension
zbMATH Keywords: isometric embedding
author: Mikhail Ostrovskii
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1371399