The Bishop-Phelps-Bollobás Theorem for bilinear forms
HTML articles powered by AMS MathViewer
- by María D. Acosta, Julio Becerra-Guerrero, Domingo García and Manuel Maestre PDF
- Trans. Amer. Math. Soc. 365 (2013), 5911-5932 Request permission
Abstract:
In this paper we provide versions of the Bishop-Phelps-Bollobás Theorem for bilinear forms. Indeed we prove the first positive result of this kind by assuming uniform convexity on the Banach spaces. A characterization of the Banach space $Y$ satisfying a version of the Bishop-Phelps-Bollobás Theorem for bilinear forms on $\ell _1 \times Y$ is also obtained. As a consequence of this characterization, we obtain positive results for finite-dimensional normed spaces, uniformly smooth spaces, the space $\mathcal {C}(K)$ of continuous functions on a compact Hausdorff topological space $K$ and the space $K(H)$ of compact operators on a Hilbert space $H$. On the other hand, the Bishop-Phelps-Bollobás Theorem for bilinear forms on $\ell _1 \times L_1 (\mu )$ fails for any infinite-dimensional $L_1 (\mu )$, a result that was known only when $L_1 (\mu ) = \ell _1$.References
- 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, DOI 10.1016/j.jfa.2008.02.014
- R. M. Aron, B. Cascales, and O. Kozhushkina, The Bishop-Phelps-Bollobás theorem and Asplund operators, Proc. Amer. Math. Soc. 139 (2011), no. 10, 3553–3560. MR 2813386, DOI 10.1090/S0002-9939-2011-10755-X
- Richard M. Aron, Yun Sung Choi, Domingo García, and Manuel Maestre, The Bishop-Phelps-Bollobás theorem for $\scr L(L_1(\mu ), L_\infty [0,1])$, Adv. Math. 228 (2011), no. 1, 617–628. MR 2822241, DOI 10.1016/j.aim.2011.05.023
- Errett Bishop and R. R. Phelps, A proof that every Banach space is subreflexive, Bull. Amer. Math. Soc. 67 (1961), 97–98. MR 123174, DOI 10.1090/S0002-9904-1961-10514-4
- Béla Bollobás, An extension to the theorem of Bishop and Phelps, Bull. London Math. Soc. 2 (1970), 181–182. MR 267380, DOI 10.1112/blms/2.2.181
- F. F. Bonsall and J. Duncan, Numerical ranges. II, London Mathematical Society Lecture Note Series, No. 10, Cambridge University Press, New York-London, 1973. MR 0442682, DOI 10.1017/CBO9780511662515
- Lixin Cheng, Duanxu Dai, and Yunbai Dong, A sharp operator version of the Bishop-Phelps theorem for operators from $\ell _1$ to CL-spaces, Proc. Amer. Math. Soc. 141 (2013), no. 3, 867–872. MR 3003679, DOI 10.1090/S0002-9939-2012-11326-7
- Yun Sung Choi and Sun Kwang Kim, The Bishop-Phelps-Bollobás property and lush spaces, J. Math. Anal. Appl. 390 (2012), no. 2, 549–555. MR 2890536, DOI 10.1016/j.jmaa.2012.01.053
- Yun Sung Choi and Hyun Gwi Song, The Bishop-Phelps-Bollobás theorem fails for bilinear forms on $l_1\times l_1$, J. Math. Anal. Appl. 360 (2009), no. 2, 752–753. MR 2561271, DOI 10.1016/j.jmaa.2009.07.008
- D. Dai, The Bishop-Phelps-Bollobás Theorem for bilinear mappings, preprint, 2011.
- Marián Fabian, Petr Habala, Petr Hájek, Vicente Montesinos, and Václav Zizler, Banach space theory, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, Springer, New York, 2011. The basis for linear and nonlinear analysis. MR 2766381, DOI 10.1007/978-1-4419-7515-7
- 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
- Miguel Martín and Rafael Payá, On CL-spaces and almost CL-spaces, Ark. Mat. 42 (2004), no. 1, 107–118. MR 2056547, DOI 10.1007/BF02432912
- Robert E. Megginson, An introduction to Banach space theory, Graduate Texts in Mathematics, vol. 183, Springer-Verlag, New York, 1998. MR 1650235, DOI 10.1007/978-1-4612-0603-3
- Rafael Payá and Yousef Saleh, New sufficient conditions for the denseness of norm attaining multilinear forms, Bull. London Math. Soc. 34 (2002), no. 2, 212–218. MR 1874249, DOI 10.1112/S0024609301008797
- W. Rudin, Real and complex analysis, McGraw-Hill Book Co., London, 1970.
- Robert Schatten, A Theory of Cross-Spaces, Annals of Mathematics Studies, No. 26, Princeton University Press, Princeton, N. J., 1950. MR 0036935
- Robert Schatten, Norm ideals of completely continuous operators, Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), Heft 27, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1960. MR 0119112, DOI 10.1007/978-3-642-87652-3
- Charles Stegall, Optimization and differentiation in Banach spaces, Proceedings of the symposium on operator theory (Athens, 1985), 1986, pp. 191–211. MR 872283, DOI 10.1016/0024-3795(86)90314-9
Additional Information
- María D. Acosta
- Affiliation: Departamento de Análisis Matemático, Universidad de Granada, Facultad de Ciencias, 18071-Granada, Spain
- Email: dacosta@ugr.es
- Julio Becerra-Guerrero
- Affiliation: Departamento de Análisis Matemático, Universidad de Granada, Facultad de Ciencias, 18071-Granada, Spain
- Email: juliobg@ugr.es
- Domingo García
- Affiliation: Departamento de Análisis Matemático, Universidad de Valencia, Doctor Moliner 50, 46100 Burjasot Valencia, Spain
- Email: domingo.garcia@uv.es
- Manuel Maestre
- Affiliation: Departamento de Análisis Matemático, Universidad de Valencia, Doctor Moliner 50, 46100 Burjasot Valencia, Spain
- Email: manuel.maestre@uv.es
- Received by editor(s): February 6, 2012
- Published electronically: July 2, 2013
- Additional Notes: The first author was supported by MICINN and FEDER Project MTM-2009–07498 and Junta de Andalucía “Proyecto de Excelencia” P09-FQM–4911
The second author was supported by Junta de Andalucía grants FQM 0199 and FQM 1215, and MTM-2011-23843
The third and fourth authors were supported by MICINN Project MTM2011-22417
The fourth author was also supported by Prometeo 2008/101 and MEC PR2010-0374. - © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 365 (2013), 5911-5932
- MSC (2010): Primary 46B20; Secondary 46B25, 46B28
- DOI: https://doi.org/10.1090/S0002-9947-2013-05881-3
- MathSciNet review: 3091270