Banach spaces with the Daugavet property

Kadets, Vladimir; Shvidkoy, Roman; Sirotkin, Gleb; Werner, Dirk

doi:10.1090/S0002-9947-99-02377-6

Banach spaces with the Daugavet property
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by Vladimir M. Kadets, Roman V. Shvidkoy, Gleb G. Sirotkin and Dirk Werner

Trans. Amer. Math. Soc. 352 (2000), 855-873

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

Published electronically: September 17, 1999

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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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