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Proceedings of the American Mathematical Society
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
Published electronically: December 6, 2012
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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.


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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: http://dx.doi.org/10.1090/S0002-9939-2012-11326-7
PII: S 0002-9939(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.