A sharp operator version of the Bishop-Phelps theorem for operators from ℓ₁ to CL-spaces

Cheng, Lixin; Dai, Duanxu; Dong, Yunbai

doi:10.1090/S0002-9939-2012-11326-7

A sharp operator version of the Bishop-Phelps theorem for operators from $\ell _1$ to CL-spaces
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Proc. Amer. Math. Soc. 141 (2013), 867-872 Request permission

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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