An exact Daugavet type inequality for small into isomorphisms in \(L_{1}\)
type: Item
instance of: scholarly article
title: An exact Daugavet type inequality for small into isomorphisms in \(L_{1}\)
zbMATH Open document ID: 1155.46004
published in: Archiv der Mathematik
publication date: 2008-07-25T00:00:00Z
review text: The Daugavet property of a Banach space \(X\) means that \(\| I + T\| = 1+ \| T\| \) for every weakly compact operator \(T : X \rightarrow X\). For \(X = L_{1}[0,1]\), the above equation in fact is true for a much wider class of operators, including the narrow operators in the sense of \textit{A.\,M.\thinspace Plichko} and \textit{M.\,M.\thinspace Popov} [Diss.\ Math.\ 306 (1990; Zbl 0715.46011)]. Developing his previous results from [Mat.\ Stud.\ 20, No.\,1, 75--84 (2003; Zbl 1056.46013)], the author shows that for every into isomorphism \(J : L_{1}[0,1] \rightarrow L_{1}[0,1]\) and for every narrow operator \(T\), \[ \| J + T \| \geq \| T \| + \| J \| \left(\tfrac{2}{d} - 1\right), \] where \(d = \| J \| \| J^{-1}\| \). It is shown that this estimate is exact.
reviewed by: Vladimir Kadets
Mathematics Subject Classification ID: 46B04
Mathematics Subject Classification ID: 47B38
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 46E30
zbMATH DE Number: 5303970
zbMATH Keywords: Daugavet equation
zbMATH Keywords: narrow operator
author: M. M. Popov
MaRDI profile type: Publication
full work available at URL: s00013-008-2559-2
OpenAlex ID: W1968019412
DOI: 10.1007/S00013-008-2559-2
description: scientific article; zbMATH DE number 5303970