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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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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