Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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

 
 

 

The Daugavet property in rearrangement invariant spaces


Authors: M. D. Acosta, A. Kamińska and M. Mastyło
Journal: Trans. Amer. Math. Soc. 367 (2015), 4061-4078
MSC (2010): Primary 46B20, 46E30
DOI: https://doi.org/10.1090/S0002-9947-2014-06166-7
Published electronically: December 3, 2014
MathSciNet review: 3324920
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study rearrangement invariant spaces with the Daugavet property. The main result of this paper states that under mild assumptions the only nonseparable rearrangement invariant space $ X$ over an atomless finite measure space with the Daugavet property is $ L_{\infty }$ endowed with its canonical norm. We also prove that a uniformly monotone rearrangement invariant space over an infinite atomless measure space with the Daugavet property is isometric to $ L_1$. As an application we obtain that an Orlicz space over an atomless measure space has the Daugavet property if and only if it is isometrically isomorphic to $ L_1$.


References [Enhancements On Off] (What's this?)

  • [1] Y. A. Abramovich and C. D. Aliprantis, An invitation to operator theory, Graduate Studies in Mathematics, vol. 50, American Mathematical Society, Providence, RI, 2002. MR 1921782 (2003h:47072)
  • [2] M. D. Acosta, A. Kamińska, and M. Mastyło, The Daugavet property and weak neighborhoods in Banach lattices, J. Convex Anal. 19 (2012), no. 3, 875-912. MR 3013764
  • [3] Colin Bennett and Robert Sharpley, Interpolation of operators, Pure and Applied Mathematics, vol. 129, Academic Press Inc., Boston, MA, 1988. MR 928802 (89e:46001)
  • [4] Shutao Chen, Geometry of Orlicz spaces, Dissertationes Math. (Rozprawy Mat.) 356 (1996), 204. With a preface by Julian Musielak. MR 1410390 (97i:46051)
  • [5] I. K. Daugavet, A property of completely continuous operators in the space $ C$, Uspehi Mat. Nauk 18 (1963), no. 5 (113), 157-158 (Russian). MR 0157225 (28 #461)
  • [6] H. Hudzik, A. Kamińska, and M. Mastyło, Monotonicity and rotundity properties in Banach lattices, Rocky Mountain J. Math. 30 (2000), no. 3, 933-950. MR 1797824 (2002c:46039), https://doi.org/10.1216/rmjm/1021477253
  • [7] Yevgen Ivakhno, Vladimir Kadets, and Dirk Werner, The Daugavet property for spaces of Lipschitz functions, Math. Scand. 101 (2007), no. 2, 261-279. MR 2379289 (2009c:46014)
  • [8] Vladimir Kadets, Miguel Martín, Javier Merí, and Dirk Werner, Lushness, numerical index 1 and the Daugavet property in rearrangement invariant spaces, Canad. J. Math. 65 (2013), no. 2, 331-348. MR 3028566, https://doi.org/10.4153/CJM-2011-096-2
  • [9] Vladimir M. Kadets, Roman V. Shvidkoy, Gleb G. Sirotkin, and Dirk Werner, Banach spaces with the Daugavet property, Trans. Amer. Math. Soc. 352 (2000), no. 2, 855-873. MR 1621757 (2000c:46023), https://doi.org/10.1090/S0002-9947-99-02377-6
  • [10] Anna Kamińska and Mieczysław Mastyło, The Dunford-Pettis property for symmetric spaces, Canad. J. Math. 52 (2000), no. 4, 789-803. MR 1767402 (2001g:46062), https://doi.org/10.4153/CJM-2000-033-9
  • [11] L. V. Kantorovich and G. P. Akilov, Functional analysis, 2nd ed., Pergamon Press, Oxford, 1982. Translated from the Russian by Howard L. Silcock. MR 664597 (83h:46002)
  • [12] S. G. Kreĭn, Yu. Ī. Petunīn, and E. M. Semënov, Interpolation of linear operators, Translations of Mathematical Monographs, vol. 54, American Mathematical Society, Providence, R.I., 1982. Translated from the Russian by J. Szűcs. MR 649411 (84j:46103)
  • [13] W. Kurc, Strictly and uniformly monotone Musielak-Orlicz spaces and applications to best approximation, J. Approx. Theory 69 (1992), no. 2, 173-187. MR 1160253 (93g:46028), https://doi.org/10.1016/0021-9045(92)90141-A
  • [14] Han Ju Lee, Monotonicity and complex convexity in Banach lattices, J. Math. Anal. Appl. 307 (2005), no. 1, 86-101. MR 2138977 (2005m:46037), https://doi.org/10.1016/j.jmaa.2005.01.017
  • [15] Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. II, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97, Springer-Verlag, Berlin, 1979. Function spaces. MR 540367 (81c:46001)
  • [16] Dirk Werner, Recent progress on the Daugavet property, Irish Math. Soc. Bull. 46 (2001), 77-97. MR 1856978 (2002i:46014)
  • [17] P. Wojtaszczyk, Some remarks on the Daugavet equation, Proc. Amer. Math. Soc. 115 (1992), no. 4, 1047-1052. MR 1126202 (92k:47041), https://doi.org/10.2307/2159353

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 46B20, 46E30

Retrieve articles in all journals with MSC (2010): 46B20, 46E30


Additional Information

M. D. Acosta
Affiliation: Departamento de Análisis Matemático, Universidad de Granada, 18071 Granada, Spain
Email: dacosta@ugr.es

A. Kamińska
Affiliation: Department of Mathematical Sciences, The University of Memphis, Memphis, Tennessee 38152
Email: kaminska@memphis.edu

M. Mastyło
Affiliation: Faculty of Mathematics and Computer Science, A. Mickiewicz University and Institute of Mathematics, Polish Academy of Sciences (Poznań branch), Umultowska 87, 61-614 Poznań, Poland
Email: mastylo@amu.edu.pl

DOI: https://doi.org/10.1090/S0002-9947-2014-06166-7
Keywords: Daugavet property, rearrangement invariant spaces, Orlicz spaces, uniform monotonicity
Received by editor(s): November 29, 2012
Received by editor(s) in revised form: March 9, 2013
Published electronically: December 3, 2014
Additional Notes: The first author was supported by MTM2012-31755, Junta de Andalucía FQM–4911 and FQM–185.
The third author was supported by the National Science Centre (NCN), Poland, grant no. 2011/01/B/ST1/06243.
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society