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The Bishop-Phelps-Bollobás theorem and Asplund operators


Authors: R. M. Aron, B. Cascales and O. Kozhushkina
Journal: Proc. Amer. Math. Soc. 139 (2011), 3553-3560
MSC (2010): Primary 46B22; Secondary 47B07
Published electronically: February 10, 2011
MathSciNet review: 2813386
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Abstract: This paper deals with a strengthening of the Bishop-Phelps property for operators that in the literature is called the Bishop-Phelps-Bollobás property. Let $ X$ be a Banach space and $ L$ a locally compact Hausdorff space. We prove that if $ T:X\to C_0(L)$ is an Asplund operator and $ \Vert T(x_0)\Vert\approxeq \Vert T\Vert$ for some $ \Vert x_0\Vert=1$, then there is a norm-attaining Asplund operator $ S:X\to C_0(L)$ and $ \Vert u_0\Vert=1$ with $ \Vert S(u_0)\Vert= \Vert S\Vert=\Vert T\Vert$ such that $ u_0\approxeq x_0$ and $ S\approxeq T$. As particular cases we obtain: (A) if $ T$ is weakly compact, then $ S$ can also be taken to be weakly compact; (B) if $ X$ is Asplund (for instance, $ X=c_0$), the pair $ (X,C_0(L))$ has the Bishop-Phelps-Bollobás property for all $ L$; (C) if $ L$ is scattered, the pair $ (X,C_0(L))$ has the Bishop-Phelps-Bollobás property for all Banach spaces $ X$.


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Additional Information

R. M. Aron
Affiliation: Department of Mathematical Sciences, Kent State University, Kent, Ohio 44242
Email: aron@math.kent.edu

B. Cascales
Affiliation: Departamento de Matemáticas, Universidad de Murcia, 30.100 Espinardo, Murcia, Spain
Email: beca@um.es

O. Kozhushkina
Affiliation: Department of Mathematical Sciences, Kent State University, Kent, Ohio 44242
Email: okozhush@math.kent.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-10755-X
Keywords: Bishop-Phelps, Bollobás, fragmentability, Asplund operator, weakly compact operator, norm-attaining
Received by editor(s): July 23, 2010
Received by editor(s) in revised form: August 24, 2010
Published electronically: February 10, 2011
Additional Notes: The research of the first author was supported in part by MICINN Project MTM2008-03211.
The research of the second author was supported by FEDER and MEC Project MTM2008- 05396 and by Fundación Séneca (CARM), project 08848/PI/08.
The research of the third author was supported in part by U.S. National Science Foundation grant DMS-0652684.
Dedicated: Dedicated to the memory of Nigel J. Kalton
Communicated by: Nigel J. Kalton
Article copyright: © Copyright 2011 American Mathematical Society