×

Almost Daugavet centers. (English) Zbl 1247.46010

A Daugavet centre is a nonzero bounded linear operator between Banach spaces, \(G: X\to Y\), such that \[ \|G+T\|= \|G\|+\|T\| \tag{1} \] for all rank-\(1\) operators \(T=x^*\otimes y\); see [T. Bosenko and V. Kadets, Zh.Mat.Fiz.Anal.Geom.6, No. 1, 3–20 (2010; Zbl 1208.46010)]. The special case \(G=\text{Id}_X\) leads, by definition, to the Daugavet property of \(X\) which is studied in many papers. On the other hand, \(X\) is said to have the almost Daugavet property if \[ \|\text{Id}+T\| = 1+\|T\| \] for “many” rank-\(1\) operators, meaning for all those \(T=x^*\otimes y\) with \(y\in X\) and \(x^*\) in some norming subspace of \(X^*\); see [V. Kadets, V. Shepelska and D. Werner, Houston J. Math.37, No. 3, 867–878 (2011; Zbl 1235.46014)].
The paper under review ties these two strands by defining an almost Daugavet centre to be a nonzero \(G:X\to Y\) such that (1) holds for all \(T=x^*\otimes y\) with \(y\in X\) and \(x^*\in G^*(Z)\) for some norming subspace \(Z\) of \(Y^*\). The author gives characterisations of almost Daugavet centres parallel to the characterisations of the almost Daugavet property in [Zbl 1235.46014]; for this he introduces the notion of thickness of an operator. He also proves that for an operator \(G\) to be an almost Daugavet centre up to renorming it is necessary and sufficient that \(G\) fixes a copy of \(\ell_1\).

MSC:

46B04 Isometric theory of Banach spaces
46B03 Isomorphic theory (including renorming) of Banach spaces
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Abramovich, Y.; Aliprantis, C. D.; Burkinshaw, O., The Daugavet equation in uniformly convex Banach spaces, J. Funct. Anal., 97, 215-230 (1991) · Zbl 0770.47005
[2] Bosenko, T.; M Kadets, V., Daugavet centers, Zh. Mat. Fiz. Anal. Geom., 6, 1, 3-20 (2010), 134 · Zbl 1208.46010
[3] R. Demazeux, Examples of almost Daugavet centers, in preparation.; R. Demazeux, Examples of almost Daugavet centers, in preparation. · Zbl 1247.46010
[4] Kadets, V. M.; Shepelska, V.; Werner, D., Thickness of the unit sphere, \( \ell_1\)-types, and the almost Daugavet property, Houston J. Math., 37, 3, 867-878 (2011) · Zbl 1235.46014
[5] Kadets, V. M.; Shvidkoy, R. V.; Sirotkin, G. G.; Werner, D., Banach spaces with the Daugavet property, Trans. Amer. Math. Soc., 352, 855-873 (2000) · Zbl 0938.46016
[6] Krivine, J.-L.; Maurey, B., Espaces de Banach stables, Israel J. Math., 39, 273-295 (1981) · Zbl 0504.46013
[7] Whitley, R., The size of the unit sphere, Canad. J. Math., 20, 450-455 (1968) · Zbl 0153.44203
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.