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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 380475, 8 pages
http://dx.doi.org/10.1155/2013/380475
Research Article

Generalized Numerical Index and Denseness of Numerical Peak Holomorphic Functions on a Banach Space

1Department of Mathematics, Kyungpook National University, Daegu 702-701, Republic of Korea
2Department of Mathematics Education, Dongguk University-Seoul, Seoul 100-715, Republic of Korea

Received 11 June 2013; Accepted 23 August 2013

Academic Editor: Geraldo Botelho

Copyright © 2013 Sung Guen Kim and Han Ju Lee. 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. S. Dineen, Complex Analysis on Infinite-Dimensional Spaces, Springer, London, UK, 1999. View at Publisher · View at Google Scholar · View at MathSciNet
  2. L. A. Harris, “The numerical range of holomorphic functions in Banach spaces,” American Journal of Mathematics, vol. 93, pp. 1005–1019, 1971. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. 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
  4. S. G. Kim, “Numerical peak points and numerical Šilov boundary for holomorphic functions,” Proceedings of the American Mathematical Society, vol. 136, no. 12, pp. 4339–4347, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. 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 MathSciNet
  6. 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 Zentralblatt MATH · View at MathSciNet · View at Scopus
  7. J. Bourgain, “On dentability and the Bishop-Phelps property,” Israel Journal of Mathematics, vol. 28, no. 4, pp. 265–271, 1977. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  8. C. Stegall, “Optimization and differentiation in Banach spaces,” Linear Algebra and Its Applications, vol. 84, pp. 191–211, 1986.
  9. J. Diestel and J. J. Uhl Jr., Vector Measures, American Mathematical Society, Providence, RI, USA, 1977. View at MathSciNet
  10. M. D. Acosta, J. Alaminos, D. García, and M. Maestre, “On holomorphic functions attaining their norms,” Journal of Mathematical Analysis and Applications, vol. 297, no. 2, pp. 625–644, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  11. Y. S. Choi, H. J. Lee, and H. G. Song, “Bishop's theorem and differentiability of a subspace of Cb(K),” Israel Journal of Mathematics, vol. 180, no. 1, pp. 93–118, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  12. M. D. Acosta and S. G. Kim, “Numerical boundaries for some classical Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 350, no. 2, pp. 694–707, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  13. E. Ed-dari, “On the numerical index of Banach spaces,” Linear Algebra and Its Applications, vol. 403, pp. 86–96, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. Y. S. Choi, D. Garcia, S. G. Kim, and M. Maestre, “The polynomial numerical index of a Banach space,” Proceedings of the Edinburgh Mathematical Society, vol. 49, no. 1, pp. 39–52, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  15. D. García, B. C. Grecu, M. Maestre, M. Martín, and J. Merí, “Two-dimensional Banach spaces with polynomial numerical index zero,” Linear Algebra and Its Applications, vol. 430, no. 8-9, pp. 2488–2500, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. S. G. Kim, “Norm and numerical radius of 2-homogeneous polynomials on the real space lp2, 1LTHEXApLTHEXA,” Kyungpook Mathematical Journal, vol. 48, no. 3, pp. 387–393, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  17. S. G. Kim, “The polynomial numerical index of Lp(μ),” Kyungpook Mathematical Journal, vol. 53, no. 1, pp. 117–124, 2013.
  18. J. Kim and H. J. Lee, “Strong peak points and strongly norm attaining points with applications to denseness and polynomial numerical indices,” Journal of Functional Analysis, vol. 257, no. 4, pp. 931–947, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  19. S. G. Kim, M. Martín, and J. Merí, “On the polynomial numerical index of the real spaces c0 and l1, l,” Journal of Mathematical Analysis and Applications, vol. 337, no. 1, pp. 98–106, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  20. H. J. Lee, “Banach spaces with polynomial numerical index 1,” Bulletin of the London Mathematical Society, vol. 40, no. 2, pp. 193–198, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  21. J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces. II, Springer, 1979. View at MathSciNet
  22. J. Johnson and J. Wolfe, “Norm attaining operators,” Polska Akademia Nauk. Institut Matematyczny. Studia Mathematica, vol. 65, no. 1, pp. 7–19, 1979. View at MathSciNet