On extension of isometries between unit spheres of \(L_{p}(\mu )\) and \(L_{p}(\nu ,H)\) (\(1< p \neq 2\), \(H\) is a Hilbert space)
type: Item
instance of: scholarly article
title: On extension of isometries between unit spheres of \(L_{p}(\mu )\) and \(L_{p}(\nu ,H)\) (\(1< p \neq 2\), \(H\) is a Hilbert space)
zbMATH Open document ID: 1112.46011
author: Xiuzhong Yang
published in: Journal of Mathematical Analysis and Applications
publication date: 2006-12-07T00:00:00Z
review text: An old and interesting question in the geometric theory of Banach spaces, due to \textit{D. Tingley} [Geom. Dedicata 22, 371--378 (1987; Zbl 0615.51005)] asks for conditions under which an onto isometry between the unit spheres extends to an affine or linear isometry between the spaces. Recently, a lot of work on problems of this kind was done by the Chinese school led by Ding Guanggui and his collaborators (see the reviews by this reviewer). In the present paper, the author introduces the notion of a \(p\)-affinely orthogonal map from an \(L_p\) space (\(p \neq 2\)) to a Banach space and shows that an \(p\)-affinely orthogonal map that is an isometry between the unit spheres extends to a linear isometry.
Mathematics Subject Classification ID: 46B04
zbMATH DE Number: 5077882
zbMATH Keywords: isometries
zbMATH Keywords: p-affinely orthogonal map
reviewed by: T.S.S.R.K. Rao
MaRDI profile type: Publication
full work available at URL: j.jmaa.2005.11.013
OpenAlex ID: W2492622366
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DOI: 10.1016/J.JMAA.2005.11.013
description: scientific article; zbMATH DE number 5077882