A sharp operator version of the BishopPhelps theorem for operators from to CLspaces
Authors:
Lixin Cheng, Duanxu Dai and Yunbai Dong
Journal:
Proc. Amer. Math. Soc. 141 (2013), 867872
MSC (2010):
Primary 47B37, 46B25; Secondary 47A58, 46B20
Published electronically:
December 6, 2012
MathSciNet review:
3003679
Fulltext PDF
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References 
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Additional Information
Abstract: Acosta et al. in 2008 gave a characterization of a Banach space (called an approximate hyperplane series property, or AHSP for short) guaranteeing exactly that a quantitative version of the BishopPhelps theorem holds for bounded operators from to the space . In this note, we give two new examples of spaces having the AHSP: the almost CLspaces and the class of Banach spaces whose dual is uniformly strongly subdifferentiable on some boundary of . We then calculate the precise parameters associated to almost CLspaces.
 1.
María
D. Acosta, Richard
M. Aron, Domingo
García, and Manuel
Maestre, The BishopPhelpsBollobás theorem for
operators, J. Funct. Anal. 254 (2008), no. 11,
2780–2799. MR 2414220
(2009c:46016), 10.1016/j.jfa.2008.02.014
 2.
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), 10.1090/S000299041961105144
 3.
Béla
Bollobás, An extension to the theorem of Bishop and
Phelps, Bull. London Math. Soc. 2 (1970),
181–182. MR 0267380
(42 #2282)
 4.
J.
Bourgain, On dentability and the BishopPhelps property,
Israel J. Math. 28 (1977), no. 4, 265–271. MR 0482076
(58 #2164)
 5.
LiXin
Cheng and Min
Li, Extreme points, exposed points,
differentiability points in CLspaces, Proc.
Amer. Math. Soc. 136 (2008), no. 7, 2445–2451. MR 2390512
(2009a:46021), 10.1090/S0002993908092204
 6.
Robert
Deville, Gilles
Godefroy, and Václav
Zizler, Smoothness and renormings in Banach spaces, Pitman
Monographs and Surveys in Pure and Applied Mathematics, vol. 64,
Longman Scientific & Technical, Harlow; copublished in the United
States with John Wiley & Sons, Inc., New York, 1993. MR 1211634
(94d:46012)
 7.
Carlo
Franchetti and Rafael
Payá, Banach spaces with strongly subdifferentiable
norm, Boll. Un. Mat. Ital. B (7) 7 (1993),
no. 1, 45–70 (English, with Italian summary). MR 1216708
(94d:46015)
 8.
R.
E. Fullerton, Geometrical characterizations of certain function
spaces, Proc. Internat. Sympos. Linear Spaces (Jerusalem, 1960)
Jerusalem Academic Press, Jerusalem; Pergamon, Oxford, 1961,
pp. 227–236. MR 0132998
(24 #A2834)
 9.
G.
Godefroy, V.
Indumathi, and F.
LustPiquard, Strong subdifferentiability of convex functionals and
proximinality, J. Approx. Theory 116 (2002),
no. 2, 397–415. MR 1911087
(2003d:41034), 10.1006/jath.2002.3679
 10.
Ȧsvald
Lima, Intersection properties of balls and
subspaces in Banach spaces, Trans. Amer. Math.
Soc. 227 (1977),
1–62. MR
0430747 (55 #3752), 10.1090/S00029947197704307474
 11.
Joram
Lindenstrauss, On operators which attain their norm, Israel J.
Math. 1 (1963), 139–148. MR 0160094
(28 #3308)
 12.
Miguel
Martín and Rafael
Payá, On CLspaces and almost CLspaces, Ark. Mat.
42 (2004), no. 1, 107–118. MR 2056547
(2005e:46019), 10.1007/BF02432912
 13.
R.
R. Phelps, The BishopPhelps theorem, Ten mathematical essays
on approximation in analysis and topology, Elsevier B. V., Amsterdam,
2005, pp. 235–244. MR 2162983
(2006d:46015), 10.1016/B9780444518613/500094
 1.
 M. D. Acosta, R. M. Aron, D. García, M. Maestre, The BishopPhelpsBollobás theorem for operators, J. Funct. Anal. 254 (2008), no. 11, 27802799. MR 2414220 (2009c:46016)
 2.
 E. Bishop and R. R. Phelps, A proof that every Banach space is subreflexive, Bull. Amer. Math. Soc. 67 (1961), 9798. MR 0123174 (23:A503)
 3.
 B. Bollobás, An extension to the theorem of Bishop and Phelps, Bull. London Math. Soc. 2 (1970), 181182. MR 0267380 (42:2282)
 4.
 J. Bourgain, On dentability and the BishopPhelps property, Israel J. Math. 28 (1977), 265271. MR 0482076 (58:2164)
 5.
 L. Cheng and M. Li, Extreme points, exposed points, differentiability points in CLspaces, Proc. Amer. Math. Soc. 136 (2008), no. 7, 24452451. MR 2390512 (2009a:46021)
 6.
 R. Deville, G. Godefroy, V. Zizler, Smoothness and renormings in Banach spaces. Pitman Monographs and Surveys in Pure and Applied Mathematics, 64. Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1993. MR 1211634 (94d:46012)
 7.
 C. Franchetti and R. Payá, Banach spaces with strongly differentiable norm, Boll. U.M.I. 7 (1993), 4570. MR 1216708 (94d:46015)
 8.
 R. E. Fullerton, Geometrical characterizations of certain function spaces, in Proc. Internat. Sympos. Linear Spaces (Jerusalem, 1960), pp. 227236, Jerusalem Academic Press, Jerusalem; Pergamon Press, Oxford, 1961. MR 0132998 (24:A2834)
 9.
 G. Godefroy, V. Indumathi, F. LustPiquard, Strong subdifferentiability of convex functionals and proximinality, J. Approx. Theory 116 (2002), 397415. MR 1911087 (2003d:41034)
 10.
 Å. Lima, Intersection properties of balls and subspaces in Banach spaces, Trans. Amer. Math. Soc. 227 (1977), 162. MR 0430747 (55:3752)
 11.
 J. Lindenstrauss, On operators which attain their norm, Israel J. Math. 1 (1963), 139148. MR 0160094 (28:3308)
 12.
 M. Martin and R. Payá, On CLspaces and almost CLspaces, Ark. Mat. 42 (2004), 107118. MR 2056547 (2005e:46019)
 13.
 R. R. Phelps, The BishopPhelps theorem, Ten mathematical essays on approximation in analysis and topology, 235244, edited by J. Ferrera, J. L pezG mez and F. R. Ruiz del Portal, Elsevier B. V., Amsterdam, 2005. MR 2162983 (2006d:46015)
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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/S000299392012113267
PII:
S 00029939(2012)113267
Keywords:
Normattaining operator,
BishopPhelps theorem,
CLspace,
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.
