Norm attaining operators into locally asymptotically midpoint uniformly convex Banach spaces

Fovelle, A.

doi:10.1090/proc/16971

Norm attaining operators into locally asymptotically midpoint uniformly convex Banach spaces
HTML articles powered by AMS MathViewer

by A. Fovelle;

Proc. Amer. Math. Soc. 152 (2024), 4835-4840

DOI: https://doi.org/10.1090/proc/16971

Published electronically: September 24, 2024

HTML | PDF | Request permission

Abstract:

We prove that if $Y$ is a locally asymptotically midpoint uniformly convex Banach space which has either a normalized, symmetric basic sequence that is not equivalent to the unit vector basis in $\ell _1$, or a normalized sequence with upper p-estimates for some $p>1$, then $Y$ does not satisfy Lindenstrauss’ property B.

References

M. D. Acosta, Denseness of norm-attaining operators into strictly convex spaces, Proc. Roy. Soc. Edinburgh Sect. A 129 (1999), no. 6, 1107–1114. MR 1728538, DOI 10.1017/S0308210500019296
María D. Acosta, Norm attaining operators into $L_1(\mu )$, Function spaces (Edwardsville, IL, 1998) Contemp. Math., vol. 232, Amer. Math. Soc., Providence, RI, 1999, pp. 1–11. MR 1678313, DOI 10.1090/conm/232/03377
María D. Acosta, Francisco J. Aguirre, and Rafael Payá, A new sufficient condition for the denseness of norm attaining operators, Rocky Mountain J. Math. 26 (1996), no. 2, 407–418. MR 1406488, DOI 10.1216/rmjm/1181072066
Francisco J. Aguirre, Norm-attaining operators into strictly convex Banach spaces, J. Math. Anal. Appl. 222 (1998), no. 2, 431–437. MR 1628476, DOI 10.1006/jmaa.1998.5913
Errett Bishop and R. R. Phelps, A proof that every Banach space is subreflexive, Bull. Amer. Math. Soc. 67 (1961), 97–98. MR 123174, DOI 10.1090/S0002-9904-1961-10514-4
S. J. Dilworth, Denka Kutzarova, N. Lovasoa Randrianarivony, J. P. Revalski, and N. V. Zhivkov, Lenses and asymptotic midpoint uniform convexity, J. Math. Anal. Appl. 436 (2016), no. 2, 810–821. MR 3446981, DOI 10.1016/j.jmaa.2015.11.061
V. Dimant, R. Gonzalo, and J. A. Jaramillo, Asymptotic structure, $l_p$-estimates of sequences, and compactness of multilinear mappings, J. Math. Anal. Appl. 350 (2009), no. 2, 680–693. MR 2474804, DOI 10.1016/j.jmaa.2008.05.046
D. J. H. Garling, On symmetric sequence spaces, Proc. London Math. Soc. (3) 16 (1966), 85–106. MR 192311, DOI 10.1112/plms/s3-16.1.85
Raquel Gonzalo, Upper and lower estimates in Banach sequence spaces, Comment. Math. Univ. Carolin. 36 (1995), no. 4, 641–653. MR 1378687
W. T. Gowers, Symmetric block bases of sequences with large average growth, Israel J. Math. 69 (1990), no. 2, 129–151. MR 1045369, DOI 10.1007/BF02937300
M. Jiménez Sevilla and Rafael Payá, Norm attaining multilinear forms and polynomials on preduals of Lorentz sequence spaces, Studia Math. 127 (1998), no. 2, 99–112. MR 1488146, DOI 10.4064/sm-127-2-99-112
Jerry Johnson and John Wolfe, Norm attaining operators, Studia Math. 65 (1979), no. 1, 7–19. MR 554537, DOI 10.4064/sm-65-1-7-19
Jerry Johnson and John Wolfe, Norm attaining operators and simultaneously continuous retractions, Proc. Amer. Math. Soc. 86 (1982), no. 4, 609–612. MR 674091, DOI 10.1090/S0002-9939-1982-0674091-6
Gilles Lancien, A short course on nonlinear geometry of Banach spaces, Topics in functional and harmonic analysis, Theta Ser. Adv. Math., vol. 14, Theta, Bucharest, 2013, pp. 77–101. MR 3184344
Joram Lindenstrauss, On operators which attain their norm, Israel J. Math. 1 (1963), 139–148. MR 160094, DOI 10.1007/BF02759700
W. L. C. Sargent, Some sequence spaces related to the $l^{p}$ spaces, J. London Math. Soc. 35 (1960), 161–171. MR 116206, DOI 10.1112/jlms/s1-35.2.161
Walter Schachermayer, Norm attaining operators on some classical Banach spaces, Pacific J. Math. 105 (1983), no. 2, 427–438. MR 691613, DOI 10.2140/pjm.1983.105.427
J. J. Uhl Jr., Norm attaining operators on $L^{1}[0,1]$ and the Radon-Nikodým property, Pacific J. Math. 63 (1976), no. 1, 293–300. MR 405076, DOI 10.2140/pjm.1976.63.293