A sharp operator version of the Bishop-Phelps theorem for operators from $\ell _1$ to CL-spaces
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- by Lixin Cheng, Duanxu Dai and Yunbai Dong PDF
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Abstract:
Acosta et al. in 2008 gave a characterization of a Banach space $Y$ (called an approximate hyperplane series property, or AHSP for short) guaranteeing exactly that a quantitative version of the Bishop-Phelps theorem holds for bounded operators from $\ell _1$ to the space $Y$. In this note, we give two new examples of spaces having the AHSP: the almost CL-spaces and the class of Banach spaces $Y$ whose dual $Y^*$ is uniformly strongly subdifferentiable on some boundary of $Y$. We then calculate the precise parameters associated to almost CL-spaces.References
- María D. Acosta, Richard M. Aron, Domingo García, and Manuel Maestre, The Bishop-Phelps-Bollobás theorem for operators, J. Funct. Anal. 254 (2008), no. 11, 2780–2799. MR 2414220, DOI 10.1016/j.jfa.2008.02.014
- 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
- Béla Bollobás, An extension to the theorem of Bishop and Phelps, Bull. London Math. Soc. 2 (1970), 181–182. MR 267380, DOI 10.1112/blms/2.2.181
- J. Bourgain, On dentability and the Bishop-Phelps property, Israel J. Math. 28 (1977), no. 4, 265–271. MR 482076, DOI 10.1007/BF02760634
- Li-Xin Cheng and Min Li, Extreme points, exposed points, differentiability points in CL-spaces, Proc. Amer. Math. Soc. 136 (2008), no. 7, 2445–2451. MR 2390512, DOI 10.1090/S0002-9939-08-09220-4
- Robert Deville, Gilles Godefroy, and Václav Zizler, Smoothness and renormings in Banach spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 64, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1993. MR 1211634
- Carlo Franchetti and Rafael Payá, Banach spaces with strongly subdifferentiable norm, Boll. Un. Mat. Ital. B (7) 7 (1993), no. 1, 45–70 (English, with Italian summary). MR 1216708
- R. E. Fullerton, Geometrical characterizations of certain function spaces, Proc. Internat. Sympos. Linear Spaces (Jerusalem, 1960) Jerusalem Academic Press, Jerusalem; Pergamon, Oxford, 1961, pp. 227–236. MR 0132998
- G. Godefroy, V. Indumathi, and F. Lust-Piquard, Strong subdifferentiability of convex functionals and proximinality, J. Approx. Theory 116 (2002), no. 2, 397–415. MR 1911087, DOI 10.1006/jath.2002.3679
- Åsvald Lima, Intersection properties of balls and subspaces in Banach spaces, Trans. Amer. Math. Soc. 227 (1977), 1–62. MR 430747, DOI 10.1090/S0002-9947-1977-0430747-4
- Joram Lindenstrauss, On operators which attain their norm, Israel J. Math. 1 (1963), 139–148. MR 160094, DOI 10.1007/BF02759700
- Miguel Martín and Rafael Payá, On CL-spaces and almost CL-spaces, Ark. Mat. 42 (2004), no. 1, 107–118. MR 2056547, DOI 10.1007/BF02432912
- R. R. Phelps, The Bishop-Phelps theorem, Ten mathematical essays on approximation in analysis and topology, Elsevier B. V., Amsterdam, 2005, pp. 235–244. MR 2162983, DOI 10.1016/B978-044451861-3/50009-4
Additional Information
- Lixin Cheng
- Affiliation: School of Mathematical Sciences, Xiamen University, Xiamen, 361005, People’s Republic of China
- Email: lxcheng@xmu.edu.cn
- Duanxu Dai
- Affiliation: School of Mathematical Sciences, Xiamen University, Xiamen, 361005, People’s Republic of China
- MR Author ID: 1003951
- Email: dduanxu@163.com
- Yunbai Dong
- Affiliation: School of Mathematical Sciences, Xiamen University, Xiamen, 361005, People’s Republic of China
- Email: Baiyunmu301@126.com
- Received by editor(s): January 21, 2011
- Received by editor(s) in revised form: June 17, 2011, June 23, 2011, June 25, 2011, and June 27, 2011
- Published electronically: December 6, 2012
- Additional Notes: The first author was supported by the Natural Science Foundation of China, grant 11771201.
- Communicated by: Thomas Schlumprecht
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 141 (2013), 867-872
- MSC (2010): Primary 47B37, 46B25; Secondary 47A58, 46B20
- DOI: https://doi.org/10.1090/S0002-9939-2012-11326-7
- MathSciNet review: 3003679