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

DOI:
https://doi.org/10.1090/S0002-9939-2011-10755-X

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 (2009c:46016)****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 (2000d:46030)****3.**Edgar Asplund,*Fréchet differentiability of convex functions*, Acta Math.**121**(1968), 31-47. MR**0231199 (37:6754)****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 (23:A503)****5.**Béla Bollobás,*An extension to the theorem of Bishop and Phelps*, Bull. London Math. Soc.**2**(1970), 181-182. MR**0267380 (42:2282)****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 (85d:46023)****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 (50:8010)****8.**J. Diestel and J. J. Uhl Jr.,*Vector measures*, Mathematical Surveys, vol. 15, American Mathematical Society, Providence, R.I., 1977, with a foreword by B. J. Pettis. MR**56:12216****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 (96i:46001)****10.**G. A. Edgar,*Asplund operators and a.e. convergence*, J. Multivariate Anal.**10**(1980), no. 3, 460-466. MR**588087 (82f:47053)****11.**Jerry Johnson and John Wolfe,*Norm attaining operators*, Studia Math.**65**(1979), no. 1, 7-19. MR**554537 (81a:47021)****12.**Joram Lindenstrauss,*On operators which attain their norm*, Israel J. Math.**1**(1963), 139-148. MR**0160094 (28:3308)****13.**I. Namioka,*Radon-Nikodým compact spaces and fragmentability*, Mathematika**34**(1987), no. 2, 258-281. MR**89i:46021****14.**I. Namioka and R. R. Phelps,*Banach spaces which are Asplund spaces*, Duke Math. J.**42**(1975), no. 4, 735-750. MR**0390721 (52:11544)****15.**A. Pełczyński and Z. Semadeni,*Spaces of continuous functions. III. Spaces for without perfect subsets*, Studia Math.**18**(1959), 211-222. MR**0107806 (21:6528)****16.**Albrecht Pietsch,*Operator ideals*, North-Holland Mathematical Library, vol. 20, North-Holland Publishing Co., Amsterdam, 1980, translated from the German by the author. MR**582655 (81j:47001)****17.**Walter Rudin,*Real and complex analysis*, third ed., McGraw-Hill Book Co., New York, 1987. MR**924157 (88k:00002)****18.**Walter Schachermayer,*Norm attaining operators and renormings of Banach spaces*, Israel J. Math.**44**(1983), no. 3, 201-212. MR**693659 (84e:46012)****19.**-,*Norm attaining operators on some classical Banach spaces*, Pacific J. Math.**105**(1983), no. 2, 427-438. MR**691613 (84g:46031)****20.**Charles Stegall,*The Radon-Nikodym property in conjugate Banach spaces*, Trans. Amer. Math. Soc.**206**(1975), 213-223. MR**0374381 (51:10581)****21.**-,*The duality between Asplund spaces and spaces with the Radon-Nikodým property*, Israel J. Math.**29**(1978), no. 4, 408-412. MR**0493268 (58:12297)****22.**-,*The Radon-Nikodým property in conjugate Banach spaces. II*, Trans. Amer. Math. Soc.**264**(1981), no. 2, 507-519. MR**603779 (82k:46030)**

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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