## Banach spaces with the Daugavet property

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- by Vladimir M. Kadets, Roman V. Shvidkoy, Gleb G. Sirotkin and Dirk Werner PDF
- Trans. Amer. Math. Soc.
**352**(2000), 855-873 Request permission

## Abstract:

A Banach space $X$ is said to have the Daugavet property if every operator $T: X\to X$ 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 $T: X\to Y$ satisfying $\|J+T\|=1+\|T\|$, where $J: X\to Y$ 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.## References

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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
- MR Author ID: 202226
- ORCID: 0000-0002-5606-2679
- 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-14 195 Berlin, Germany
- Email: werner@math.fu-berlin.de
- 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. - © Copyright 1999 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**352**(2000), 855-873 - MSC (1991): Primary 46B20; Secondary 46B04, 47B38
- DOI: https://doi.org/10.1090/S0002-9947-99-02377-6
- MathSciNet review: 1621757