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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Banach spaces with the Daugavet property


Authors: Vladimir M. Kadets, Roman V. Shvidkoy, Gleb G. Sirotkin and Dirk Werner
Journal: Trans. Amer. Math. Soc. 352 (2000), 855-873
MSC (1991): Primary 46B20; Secondary 46B04, 47B38
Published electronically: September 17, 1999
MathSciNet review: 1621757
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Abstract: A Banach space $X$ is said to have the Daugavet property if every operator $T:\allowbreak X\to X$ of rank $1$ satisfies $\|\operatorname{Id}+T\| = 1+\|T\|$. We show that then every weakly compact operator satisfies this equation as well and that $X$ contains a copy of $\ell _{1}$. However, $X$ 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.


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Additional Information

Vladimir M. Kadets
Affiliation: Faculty of Mechanics and Mathematics, Kharkov State University, pl. Svobody 4 310077 Kharkov, Ukraine
Address at time of publication: I. Mathematisches Institut, Freie Universität Berlin, Arnimallee 2–6, D-14195 Berlin, Germany
Email: kadets@math.fu-berlin.de

Roman V. Shvidkoy
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
Email: shvidkoy_r@yahoo.com

Gleb G. Sirotkin
Affiliation: Department of Mathematics, Indiana University-Purdue University Indianapolis, 402 Blackford Street, Indianapolis, Indiana 46202

Dirk Werner
Affiliation: I. Mathematisches Institut, Freie Universität Berlin, Arnimallee 2–6, D-14195 Berlin, Germany
Email: werner@math.fu-berlin.de

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02377-6
PII: S 0002-9947(99)02377-6
Keywords: Daugavet equation, Daugavet property, unconditional bases
Received by editor(s): October 6, 1997
Published electronically: September 17, 1999
Additional Notes: The work of the first-named author was done during his visit to Freie Universität Berlin, where he was supported by a grant from the Deutscher Akademischer Austauschdienst. He was also supported by INTAS grant 93-1376.
Article copyright: © Copyright 1999 American Mathematical Society