The Daugavet property in rearrangement invariant spaces

Acosta, M.; Kamińska, A.; Mastyło, M.

doi:10.1090/S0002-9947-2014-06166-7

The Daugavet property in rearrangement invariant spaces
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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