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Abstract and Applied Analysis
Volume 2014, Article ID 479208, 15 pages
http://dx.doi.org/10.1155/2014/479208
Research Article

On the Bishop-Phelps-Bollobás Property for Numerical Radius

1Department of Mathematics, Kyonggi University, Suwon 443-760, Republic of Korea
2Department of Mathematics Education, Dongguk University-Seoul, Seoul 100-715, Republic of Korea
3Departamento de Análisis Mátematico, Facultad de Ciencias, Universidad de Granada, E-18071 Granada, Spain

Received 30 December 2013; Accepted 14 February 2014; Published 6 April 2014

Academic Editor: Manuel Maestre

Copyright © 2014 Sun Kwang Kim et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. F. F. Bonsall and J. Duncan, Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras, vol. 2 of London Mathematical Society Lecture Note Series 2, Cambridge University Press, London, UK, 1971. View at MathSciNet
  2. F. F. Bonsall and J. Duncan, Numerical Ranges II, Cambridge University Press, London, UK, 1973, London Mathematical Society Lecture Notes Series 10. View at MathSciNet
  3. B. Sims, On numerical range and its applications to Banach algebras [Ph.D. thesis], University of Newcastle, New South Wales, Australia, 1972.
  4. I. D. Berg and B. Sims, “Denseness of operators which attain their numerical radius,” Australian Mathematical Society A, vol. 36, no. 1, pp. 130–133, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. C. S. Cardassi, “Density of numerical radius attaining operators on some reflexive spaces,” Bulletin of the Australian Mathematical Society, vol. 31, no. 1, pp. 1–3, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. C. S. Cardassi, “Numerical radius attaining operators,” in Banach Spaces (Columbia, Mo., 1984), vol. 1166 of Lecture Notes in Mathematics, pp. 11–14, Springer, Berlin, Germany, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. C. S. Cardassi, “Numerical radius-attaining operators on C(K),” Proceedings of the American Mathematical Society, vol. 95, no. 4, pp. 537–543, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. M. D. Acosta, Operadores que alcanzan su radio numerico [Ph.D. thesis], Universidad de Granada, Granada, Spain, 1990.
  9. M. D. Acosta and R. Payá, “Numerical radius attaining operators and the Radon-Nikodým property,” The Bulletin of the London Mathematical Society, vol. 25, no. 1, pp. 67–73, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. R. Payá, “A counterexample on numerical radius attaining operators,” Israel Journal of Mathematics, vol. 79, no. 1, pp. 83–101, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. Y. S. Choi and S. G. Kim, “Norm or numerical radius attaining multilinear mappings and polynomials,” Journal of the London Mathematical Society, vol. 54, no. 1, pp. 135–147, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. M. D. Acosta, J. Becerra Guerrero, and M. Ruiz Galán, “Numerical-radius-attaining polynomials,” The Quarterly Journal of Mathematics, vol. 54, no. 1, pp. 1–10, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. M. D. Acosta and S. G. Kim, “Denseness of holomorphic functions attaining their numerical radii,” Israel Journal of Mathematics, vol. 161, pp. 373–386, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. S. G. Kim and H. J. Lee, “Numerical peak holomorphic functions on Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 364, no. 2, pp. 437–452, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. M. D. Acosta, R. M. Aron, D. García, and M. Maestre, “The Bishop-Phelps-Bollobás theorem for operators,” Journal of Functional Analysis, vol. 254, no. 11, pp. 2780–2799, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. A. J. Guirao and O. Kozhushkina, “The Bishop-Phelps-Bollobás property for numerical radius in 1(C),” Studia Mathematica, vol. 218, no. 1, pp. 41–54, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. J. Falcó, “The Bishop-Phelps-Bollobas property for numerical radius on L1,” Journal of Mathematical Analysis and Applications, vol. 414, no. 1, pp. 125–133, 2014. View at Google Scholar
  18. A. Aviles, A. J. Guirao, and J. Rodriguez, “On the Bishop-Phelps-Bollobas property for numerical radius in CK-spaces,” Preprint.
  19. R. M. Aron, Y. S. Choi, S. K. Kim, H. J. Lee, and M. Martin, “The Bishop-Phelps-Bollobas version of Lindenstrauss properties A and B,” Preprint.
  20. V. Kadets, M. Martín, and R. Payá, “Recent progress and open questions on the numerical index of Banach spaces,” Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales A, vol. 100, no. 1-2, pp. 155–182, 2006. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. M. Martín, J. Merí, and M. Popov, “On the numerical index of real Lp(μ)-spaces,” Israel Journal of Mathematics, vol. 184, pp. 183–192, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  22. A. Iwanik, “Norm attaining operators on Lebesgue spaces,” Pacific Journal of Mathematics, vol. 83, no. 2, pp. 381–386, 1979. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. H. H. Schaefer, Banach Lattices and Positive Operators, Springer, New York, NY, USA, 1974. View at MathSciNet
  24. J. Duncan, C. M. McGregor, J. D. Pryce, and A. J. White, “The numerical index of a normed space,” Journal of the London Mathematical Society. Second Series, vol. 2, pp. 481–488, 1970. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. Y. S. Choi, S. K. Kim, H. J. Lee, and M. Martin, “The Bishop-Phelps-Bollobas theorem for operators on L1μ,” Preprint.
  26. H. E. Lacey, The Isometric Theory of Classical Banach Spaces, Springer, New York, NY, USA, 1974. View at MathSciNet
  27. G. Godefroy, “Universal spaces for strictly convex Banach spaces,” Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales A, vol. 100, no. 1-2, pp. 137–146, 2006. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. C. Finet, M. Martín, and R. Payá, “Numerical index and renorming,” Proceedings of the American Mathematical Society, vol. 131, no. 3, pp. 871–877, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet