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Banach spaces with the Daugavet property

Authors:	Vladimir M Kadets Roman V Shvidkoy Gleb G Sirotkin Dirk Werner

Institution:	Faculty of Mechanics and Mathematics, Kharkov State University, pl. Svobody 4, 310077 Kharkov, Ukraine ; Department of Mathematics, University of Missouri, Columbia, Missouri 65211 ; Department of Mathematics, Indiana University-Purdue University Indianapolis, 402 Blackford Street, Indianapolis, Indiana 46202 Dirk Werner ; I. Mathematisches Institut, Freie Universität Berlin, Arnimallee 2--6, D-14195 Berlin, Germany

Abstract:	A Banach space is said to have the Daugavet property if every operator $T:\allowbreak X\to X$ of rank satisfies $\\|\operatorname{Id}+T\\| = 1+\\|T\\|$ . We show that then every weakly compact operator satisfies this equation as well and that contains a copy of $\ell _{1}$ . However, need not contain a copy of $L_{1}$ . We also study pairs of spaces $X\subset Y$ and operators $T:\allowbreak X\to Y$ satisfying $\\|J+T\\|=1+\\|T\\|$ , where $J:\allowbreak X\to Y$ is the natural embedding. This leads to the result that a Banach space with the Daugavet property does not embed into a space with an unconditional basis. In another direction, we investigate spaces where the set of operators with $\\|\operatorname{Id}+T\\|=1+\\|T\\|$ is as small as possible and give characterisations in terms of a smoothness condition.

Keywords:	Daugavet equation Daugavet property unconditional bases

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