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Transactions of the American Mathematical Society

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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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  • 1. Y. Abramovich. New classes of spaces on which compact operators satisfy the Daugavet equation. J. Operator Theory 25 (1991), 331-345. MR 94a:47061
  • 2. Y. Abramovich, C. D. Aliprantis, and O. Burkinshaw. The Daugavet equation in uniformly convex Banach spaces. J. Funct. Anal. 97 (1991), 215-230. MR 92i:47005
  • 3. V. F. Babenko and S. A. Pichugov. On a property of compact operators in the space of integrable functions. Ukrainian Math. J. 33 (1981), 374-376. MR 82m:47018
  • 4. B. Beauzamy. Introduction to Banach Spaces and their Geometry. North-Holland, Amsterdam-New York-Oxford, second edition, 1985. MR 88f:46021
  • 5. E. Behrends. On the principle of local reflexivity. Studia Math. 100 (1991), 109-128. MR 92j:46021
  • 6. J. Bourgain. Strongly exposed points in weakly compact convex sets in Banach spaces. Proc. Amer. Math. Soc. 58 (1976), 197-200. MR 54:3363
  • 7. G. Choquet. Lectures on Analysis, Vol. II. W. A. Benjamin, New York, 1969. MR 40:3253
  • 8. I. K. Daugavet. On a property of completely continuous operators in the space $C$. Uspekhi Mat. Nauk 18.5 (1963), 157-158 (Russian). MR 28:461
  • 9. J. Diestel. Geometry of Banach Spaces - Selected Topics. Lecture Notes in Math. 485. Springer, Berlin-Heidelberg-New York, 1975. MR 57:1079
  • 10. P. N. Dowling, W. B. Johnson, C. J. Lennard, and B. Turett. The optimality of James's distortion theorem. Proc. Amer. Math. Soc. 125 (1997), 167-174. MR 97d:46010
  • 11. C. Foia\c{s} and I. Singer. Points of diffusion of linear operators and almost diffuse operators in spaces of continuous functions. Math. Z. 87 (1965), 434-450. MR 31:5093
  • 12. P. Habala, P. Hájek, and V. Zizler. Introduction to Banach Spaces. Matfyz Press, Prague, 1996.
  • 13. P. Harmand, D. Werner, and W. Werner. $M$-Ideals in Banach Spaces and Banach Algebras. Lecture Notes in Math. 1547. Springer, Berlin-Heidelberg-New York, 1993. MR 94k:46022
  • 14. J. R. Holub. Daugavet's equation and operators on $L_1(\mu)$. Proc. Amer. Math. Soc. 100 (1987), 295-300. MR 88j:47037
  • 15. V. M. Kadets. Two-dimensional universal Banach spaces. C. R. Acad. Bulgare Sci. 35 (1982), 1331-1332 (Russian). MR 86b:46024
  • 16. V. M. Kadets. Some remarks concerning the Daugavet equation. Quaestiones Math. 19 (1996), 225-235. MR 97c:46015
  • 17. V. M. Kadets and M. M. Popov. The Daugavet property for narrow operators in rich subspaces of $C[0,1]$ and $L_{1}[0,1]$. Algebra i Analiz. 8 (1996), 43-62; English transl., St. Petersburg Math. J. 8 (1997), 521-584. MR 97f:47030
  • 18. V. M. Kadets and R. V. Shvidkoy. The Daugavet property for pairs of Banach spaces. Math. Analysis, Algebra and Geometry (to appear).
  • 19. V. M. Kadets, R. V. Shvidkoy, G. G. Sirotkin and D. Werner. Espaces de Banach ayant la propriété de Daugavet. C. R. Acad. Sc. Paris, Sér. I, 325 (1997), 1291-1294. MR 98i:46011
  • 20. J. L. Kelley. General Topology. Van Nostrand, 1955. MR 16:1136c
  • 21. G. Ya. Lozanovskii. On almost integral operators in KB-spaces. Vestnik Leningrad Univ. Mat. Mekh. Astr. 21.7 (1966), 35-44 (Russian). MR 34:8185
  • 22. I. Nazarenko. Paper to appear.
  • 23. A. M. Plichko and M. M. Popov. Symmetric function spaces on atomless probability spaces. Dissertationes Mathematicae 306 (1990). MR 92f:46032
  • 24. M. Talagrand. The three-space problem for $L^1$. J. Amer. Math. Soc. 3 (1990), 9-29. MR 90i:46059
  • 25. L. Weis. Approximation by weakly compact operators on $L_{1}$. Math. Nachr. 119 (1984), 321-326. MR 86h:47070
  • 26. L. Weis and D. Werner. The Daugavet equation for operators not fixing a copy of $C[0,1]$. J. Operator Theory 39 (1998), 89-98. MR 99b:47049
  • 27. D. Werner. The Daugavet equation for operators on function spaces. J. Funct. Anal. 143 (1997), 117-128. MR 98c:47025
  • 28. P. Wojtaszczyk. Some remarks on the Daugavet equation. Proc. Amer. Math. Soc. 115 (1992), 1047-1052. MR 92k:47041

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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

Roman V. Shvidkoy
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211

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

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

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