The polynomial Daugavet property for atomless \(L_{1}(\mu)\)-spaces
type: Item
instance of: scholarly article
title: The polynomial Daugavet property for atomless \(L_{1}(\mu)\)-spaces
zbMATH Open document ID: 1201.46012
author: Miguel Martín
author: Javier Merí
author: M. M. Popov
published in: Archiv der Mathematik
publication date: 2010-05-17T00:00:00Z
review text: The (linear) Daugavet property of a Banach space \(X\) can be defined by saying that \(\|\text{Id}+T\|=1+\|T\|\) for each weakly compact linear operator \(T:X\to X\). Likewise, the polynomial Daugavet property requires that \(\|\text{Id}+P\|=1+\|P\|\) for each weakly compact polynomial \(P: X\to X\). The classical examples of Banach spaces with the Daugavet property are vector-valued \(C(K)\)- and \(L_1\)-spaces, where \(K\) is perfect and the measure is atomless, and it has been proved in [\textit{Y.\,S.\thinspace Choi, D.\,García, M.\,Maestre} and \textit{M.\,Martín}, Stud.\ Math.\ 178, No.\,1, 63--82 (2007; Zbl 1121.46039)] that those \(C(K,E)\)-spaces also have the polynomial Daugavet property. Here, the authors extend this result to atomless \(L_1(\mu,E)\)-spaces, for which a different technique of proof is needed. They also pose the problem to decide whether the Daugavet property implies the polynomial Daugavet property in general.
reviewed by: Dirk Werner
Mathematics Subject Classification ID: 46B04
Mathematics Subject Classification ID: 46G25
Mathematics Subject Classification ID: 46E40
zbMATH DE Number: 5708651
zbMATH Keywords: Daugavet property
zbMATH Keywords: spaces of polynomials
zbMATH Keywords: Bochner \(L_1\)-spaces
MaRDI profile type: Publication
full work available at URL: s00013-010-0105-5
OpenAlex ID: W2049303594
cites work: Q5317928
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cites work: Q5513421
cites work: Banach spaces with the Daugavet property
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DOI: 10.1007/S00013-010-0105-5
description: scientific article; zbMATH DE number 5708651