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Some Examples in Classical Banach Spaces

  • Vladimir Kadets
  • Miguel Martín
  • Javier Merí
  • Antonio Pérez
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2205)

Abstract

Our aim here is to present examples of operators which are lush, spear, or have the aDP, defined in some classical Banach spaces. One of the most intriguing examples is the Fourier transform on L1, which we prove that is lush. Next, we study a number of examples of operators arriving to spaces of continuous functions. In particular, it is shown that every uniform algebra is lush-embedded into a space of bounded continuous functions. Finally, examples of operators acting from spaces of integrable functions are studied.

References

  1. 8.
    R. Aron, B. Cascales, O. Kozhushkina, The Bishop-Phelps-Bollobás theorem and Asplund operators. Proc. Am. Math. Soc. 139, 3553–3560 (2011)CrossRefGoogle Scholar
  2. 21.
    K. Boyko, V. Kadets, M. Martín, D. Werner, Numerical index of Banach spaces and duality. Math. Proc. Camb. 142, 93–102 (2007)MathSciNetCrossRefGoogle Scholar
  3. 24.
    B. Cascales, V. Kadets, J. Rodríguez, Measurability and selections of multi-functions in Banach spaces. J. Convex Anal. 17, 229–240 (2010)MathSciNetzbMATHGoogle Scholar
  4. 25.
    B. Cascales, A. Guirao, V. Kadets, A Bishop–Phelps–Bollobás type theorem for uniform algebras. Adv. Math. 240, 370–382 (2013)MathSciNetCrossRefGoogle Scholar
  5. 26.
    P. Cembranos, J. Mendoza, Banach Spaces of Vector-Valued Functions. Lecture Notes in Mathematics, vol. 1676 (Springer, Berlin, 1997)Google Scholar
  6. 31.
    H. Dales, Banach Algebras and Automatic Continuity (Clarendon Press, Oxford, 2000)zbMATHGoogle Scholar
  7. 36.
    J. Diestel, J. Uhl, Vector Measures. Mathematical Surveys, No. 15 (AMS, Providence, RI, 1977), XIIIGoogle Scholar
  8. 38.
    M. Fabian, P. Habala, P. Hájek, V. Montesinos, V. Zizler, Banach Space Theory: The Basis for Linear and Nonlinear Analysis (Springer Science and Business Media, New York, 2011)CrossRefGoogle Scholar
  9. 42.
    T. Gamelin, Uniform Algebras, 2nd edn. (AMS Chelsea Publishing, Providence, RI, 2005)zbMATHGoogle Scholar
  10. 53.
    P. Harmand, D. Werner, D. Werner, M-Ideals in Banach Spaces and Banach Algebras. Lecture Notes in Mathematics, vol. 1547 (Springer, Berlin, 1993)Google Scholar
  11. 59.
    V. Kadets, M. Popov, The Daugavet property for narrow operators in rich subspaces of C[0, 1] and L 1[0, 1]. St. Petersburg Math. J. 8, 571–584 (1997)Google Scholar
  12. 63.
    V. Kadets, N. Kalton, D. Werner, Remarks on rich subspaces of Banach spaces. Stud. Math. 159, 195–206 (2003)MathSciNetCrossRefGoogle Scholar
  13. 67.
    V. Kadets, M. Martín, J. Merí, V. Shepelska, Lushness, numerical index one and duality. J. Math. Anal. Appl. 357, 15–24 (2009)MathSciNetCrossRefGoogle Scholar
  14. 85.
    S. Luecking, The Daugavet property and translation-invariant subspaces. Stud. Math. 221, 269–291 (2014)MathSciNetCrossRefGoogle Scholar
  15. 110.
    W. Rudin, Fourier Analysis on Groups (Wiley, New York, 1990)CrossRefGoogle Scholar
  16. 123.
    D. Werner, The Daugavet equation for operators on function spaces. J. Funct. Anal. 143, 117–128 (1997)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Vladimir Kadets
    • 1
  • Miguel Martín
    • 2
  • Javier Merí
    • 2
  • Antonio Pérez
    • 3
  1. 1.School of Mathematics and Computer ScienceV. N. Karazin Kharkiv National UniversityKharkivUkraine
  2. 2.Departamento de Análisis MatemáticoUniversidad de GranadaGranadaSpain
  3. 3.Departamento de MatemáticasUniversidad de MurciaMurciaSpain

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