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

DOI:
https://doi.org/10.1090/S0002-9947-99-02377-6

Published electronically:
September 17, 1999

MathSciNet review:
1621757

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Abstract | References | Similar Articles | Additional Information

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.

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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

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