The Daugavet property in rearrangement invariant spaces
HTML articles powered by AMS MathViewer
- by M. D. Acosta, A. Kamińska and M. Mastyło PDF
- Trans. Amer. Math. Soc. 367 (2015), 4061-4078 Request permission
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
- 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, DOI 10.1090/gsm/050
- 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
- Colin Bennett and Robert Sharpley, Interpolation of operators, Pure and Applied Mathematics, vol. 129, Academic Press, Inc., Boston, MA, 1988. MR 928802
- Shutao Chen, Geometry of Orlicz spaces, Dissertationes Math. (Rozprawy Mat.) 356 (1996), 204. With a preface by Julian Musielak. MR 1410390
- 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
- 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, DOI 10.1216/rmjm/1021477253
- 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, DOI 10.7146/math.scand.a-15044
- 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, DOI 10.4153/CJM-2011-096-2
- 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, DOI 10.1090/S0002-9947-99-02377-6
- Anna Kamińska and Mieczysław Mastyło, The Dunford-Pettis property for symmetric spaces, Canad. J. Math. 52 (2000), no. 4, 789–803. MR 1767402, DOI 10.4153/CJM-2000-033-9
- L. V. Kantorovich and G. P. Akilov, Functional analysis, 2nd ed., Pergamon Press, Oxford-Elmsford, N.Y., 1982. Translated from the Russian by Howard L. Silcock. MR 664597
- 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
- 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, DOI 10.1016/0021-9045(92)90141-A
- Han Ju Lee, Monotonicity and complex convexity in Banach lattices, J. Math. Anal. Appl. 307 (2005), no. 1, 86–101. MR 2138977, DOI 10.1016/j.jmaa.2005.01.017
- 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-New York, 1979. Function spaces. MR 540367
- Dirk Werner, Recent progress on the Daugavet property, Irish Math. Soc. Bull. 46 (2001), 77–97. MR 1856978
- P. Wojtaszczyk, Some remarks on the Daugavet equation, Proc. Amer. Math. Soc. 115 (1992), no. 4, 1047–1052. MR 1126202, DOI 10.1090/S0002-9939-1992-1126202-2
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
- MR Author ID: 121145
- Email: mastylo@amu.edu.pl
- 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. - © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 3324920