Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A sharp operator version of the Bishop-Phelps theorem for operators from $ \ell_1$ to CL-spaces


Authors: Lixin Cheng, Duanxu Dai and Yunbai Dong
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
Published electronically: December 6, 2012
MathSciNet review: 3003679
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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

  • 1. M. D. Acosta, R. M. Aron, D. García, M. Maestre, The Bishop-Phelps-Bollobás theorem for operators, J. Funct. Anal. 254 (2008), no. 11, 2780-2799. MR 2414220 (2009c:46016)
  • 2. E. Bishop and R. R. Phelps, A proof that every Banach space is subreflexive, Bull. Amer. Math. Soc. 67 (1961), 97-98. MR 0123174 (23:A503)
  • 3. B. Bollobás, An extension to the theorem of Bishop and Phelps, Bull. London Math. Soc. 2 (1970), 181-182. MR 0267380 (42:2282)
  • 4. J. Bourgain, On dentability and the Bishop-Phelps property, Israel J. Math. 28 (1977), 265-271. MR 0482076 (58:2164)
  • 5. L. Cheng and M. Li, Extreme points, exposed points, differentiability points in CL-spaces, Proc. Amer. Math. Soc. 136 (2008), no. 7, 2445-2451. MR 2390512 (2009a:46021)
  • 6. R. Deville, G. Godefroy, V. Zizler, Smoothness and renormings in Banach spaces. Pitman Monographs and Surveys in Pure and Applied Mathematics, 64. Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1993. MR 1211634 (94d:46012)
  • 7. C. Franchetti and R. Payá, Banach spaces with strongly differentiable norm, Boll. U.M.I. 7 (1993), 45-70. MR 1216708 (94d:46015)
  • 8. R. E. Fullerton, Geometrical characterizations of certain function spaces, in Proc. Internat. Sympos. Linear Spaces (Jerusalem, 1960), pp. 227-236, Jerusalem Academic Press, Jerusalem; Pergamon Press, Oxford, 1961. MR 0132998 (24:A2834)
  • 9. G. Godefroy, V. Indumathi, F. Lust-Piquard, Strong subdifferentiability of convex functionals and proximinality, J. Approx. Theory 116 (2002), 397-415. MR 1911087 (2003d:41034)
  • 10. Å. Lima, Intersection properties of balls and subspaces in Banach spaces, Trans. Amer. Math. Soc. 227 (1977), 1-62. MR 0430747 (55:3752)
  • 11. J. Lindenstrauss, On operators which attain their norm, Israel J. Math. 1 (1963), 139-148. MR 0160094 (28:3308)
  • 12. M. Martin and R. Payá, On CL-spaces and almost CL-spaces, Ark. Mat. 42 (2004), 107-118. MR 2056547 (2005e:46019)
  • 13. R. R. Phelps, The Bishop-Phelps theorem, Ten mathematical essays on approximation in analysis and topology, 235-244, edited by J. Ferrera, J. L $ \acute {\textrm {o}}$pez-G $ \acute {\rm {o}}$mez and F. R. Ruiz del Portal, Elsevier B. V., Amsterdam, 2005. MR 2162983 (2006d:46015)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 47B37, 46B25, 47A58, 46B20

Retrieve articles in all journals with MSC (2010): 47B37, 46B25, 47A58, 46B20


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
Email: dduanxu@163.com

Yunbai Dong
Affiliation: School of Mathematical Sciences, Xiamen University, Xiamen, 361005, People’s Republic of China
Email: Baiyunmu301@126.com

DOI: https://doi.org/10.1090/S0002-9939-2012-11326-7
Keywords: Norm-attaining operator, Bishop-Phelps theorem, CL-space, Banach space
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
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society