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 | References | Similar Articles | Additional Information

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 be a Banach space and a locally compact Hausdorff space. We prove that if is an Asplund operator and for some , then there is a norm-attaining Asplund operator and with such that and . As particular cases we obtain: (A) if is weakly compact, then can also be taken to be weakly compact; (B) if is Asplund (for instance, ), the pair has the Bishop-Phelps-Bollobás property for all ; (C) if is scattered, the pair has the Bishop-Phelps-Bollobás property for all Banach spaces .

**1.**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**, 10.1016/j.jfa.2008.02.014**2.**J. Alaminos, Y. S. Choi, S. G. Kim, and R. Payá,*Norm attaining bilinear forms on spaces of continuous functions*, Glasgow Math. J.**40**(1998), no. 3, 359–365. MR**1660038**, 10.1017/S0017089500032717**3.**Edgar Asplund,*Fréchet differentiability of convex functions*, Acta Math.**121**(1968), 31–47. MR**0231199****4.**Errett Bishop and R. R. Phelps,*A proof that every Banach space is subreflexive*, Bull. Amer. Math. Soc.**67**(1961), 97–98. MR**0123174**, 10.1090/S0002-9904-1961-10514-4**5.**Béla Bollobás,*An extension to the theorem of Bishop and Phelps*, Bull. London Math. Soc.**2**(1970), 181–182. MR**0267380****6.**Richard D. Bourgin,*Geometric aspects of convex sets with the Radon-Nikodým property*, Lecture Notes in Mathematics, vol. 993, Springer-Verlag, Berlin, 1983. MR**704815****7.**W. J. Davis, T. Figiel, W. B. Johnson, and A. Pełczyński,*Factoring weakly compact operators*, J. Functional Analysis**17**(1974), 311–327. MR**0355536****8.**J. Diestel and J. J. Uhl Jr.,*Vector measures*, American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis; Mathematical Surveys, No. 15. MR**0453964****9.**Joe Diestel, Hans Jarchow, and Andrew Tonge,*Absolutely summing operators*, Cambridge Studies in Advanced Mathematics, vol. 43, Cambridge University Press, Cambridge, 1995. MR**1342297****10.**G. A. Edgar,*Asplund operators and a.e. convergence*, J. Multivariate Anal.**10**(1980), no. 3, 460–466. MR**588087**, 10.1016/0047-259X(80)90064-0**11.**Jerry Johnson and John Wolfe,*Norm attaining operators*, Studia Math.**65**(1979), no. 1, 7–19. MR**554537****12.**Joram Lindenstrauss,*On operators which attain their norm*, Israel J. Math.**1**(1963), 139–148. MR**0160094****13.**I. Namioka,*Radon-Nikodým compact spaces and fragmentability*, Mathematika**34**(1987), no. 2, 258–281. MR**933504**, 10.1112/S0025579300013504**14.**I. Namioka and R. R. Phelps,*Banach spaces which are Asplund spaces*, Duke Math. J.**42**(1975), no. 4, 735–750. MR**0390721****15.**A. Pełczyński and Z. Semadeni,*Spaces of continuous functions. III. Spaces 𝐶(Ω) for Ω without perfect subsets*, Studia Math.**18**(1959), 211–222. MR**0107806****16.**Albrecht Pietsch,*Operator ideals*, North-Holland Mathematical Library, vol. 20, North-Holland Publishing Co., Amsterdam-New York, 1980. Translated from German by the author. MR**582655****17.**Walter Rudin,*Real and complex analysis*, 3rd ed., McGraw-Hill Book Co., New York, 1987. MR**924157****18.**Walter Schachermayer,*Norm attaining operators and renormings of Banach spaces*, Israel J. Math.**44**(1983), no. 3, 201–212. MR**693659**, 10.1007/BF02760971**19.**Walter Schachermayer,*Norm attaining operators on some classical Banach spaces*, Pacific J. Math.**105**(1983), no. 2, 427–438. MR**691613****20.**Charles Stegall,*The Radon-Nikodym property in conjugate Banach spaces*, Trans. Amer. Math. Soc.**206**(1975), 213–223. MR**0374381**, 10.1090/S0002-9947-1975-0374381-1**21.**Charles Stegall,*The duality between Asplund spaces and spaces with the Radon-Nikodým property*, Israel J. Math.**29**(1978), no. 4, 408–412. MR**0493268****22.**Charles Stegall,*The Radon-Nikodým property in conjugate Banach spaces. II*, Trans. Amer. Math. Soc.**264**(1981), no. 2, 507–519. MR**603779**, 10.1090/S0002-9947-1981-0603779-1

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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:
https://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