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The Bishop–Phelps–Bollobás Property and Absolute Sums

  • Yun Sung Choi
  • Sheldon Dantas
  • Mingu JungEmail author
  • Miguel Martín
Article

Abstract

In this paper, we study conditions assuring that the Bishop–Phelps–Bollobás property (BPBp, for short) is inherited by absolute summands of the range space or of the domain space. Concretely, given a pair (XY) of Banach spaces having the BPBp,
  1. (a)

    if \(Y_1\) is an absolute summand of Y, then \((X,Y_1)\) has the BPBp;

     
  2. (b)

    if \(X_1\) is an absolute summand of X of type 1 or \(\infty \), then \((X_1,Y)\) has the BPBp.

     
Besides, analogous results for the BPBp for compact operators and for the density of norm-attaining operators are also given. We also show that the Bishop–Phelps–Bollobás property for numerical radius is inherited by absolute summands of type 1 or \(\infty \). Moreover, we provide analogous results for numerical radius attaining operators and for the BPBp for numerical radius for compact operators.

Keywords

Bishop–Phelps theorem Bishop–Phelps–Bollobás property Norm-attaining operators Absolute sums 

Mathematics Subject Classification

Primary 46B04 Secondary 46B20 46E40 47A12 

Notes

References

  1. 1.
    Acosta, M.D.: Denseness of norm attaining mappings. RACSAM 100, 9–30 (2006)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Acosta, M.D., Aron, R.M., García, D., Maestre, M.: The Bishop–Phelps–Bollobás theorem for operators. J. Funct. Anal. 294, 2780–2899 (2008)CrossRefGoogle Scholar
  3. 3.
    Acosta, M.D., Fakhar, M., Soleimani-Mourchehkhorti, M.: The Bishop–Phelps–Bollobás property for numerical radius of operators on \(L_1(\mu )\). J. Math. Anal. Appl. 458, 925–936 (2018)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Acosta, M.D., Mastyło, M., Soleimani-Mourchehkhorti, M.: The Bishop–Phelps–Bollobás and approximate hyperplane series properties. J. Funct. Anal. 274(9), 2673–2699 (2018)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Aron, R.M., Choi, Y.S., Kim, S.K., Lee, H.J., Martín, M.: The Bishop–Phelps–Bollobás version of Lindenstrauss properties A and B. Trans. Am. Math. Soc. 367, 6085–6101 (2015)CrossRefGoogle Scholar
  6. 6.
    Avilés, A., Guirao, A.J., Rodríguez, J.: On the Bishop–Phelps–Bollobás property for numerical radius in \(C(K)\) spaces. J. Math. Anal. Appl. 419, 395–421 (2014)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bishop, E., Phelps, R.R.: A proof that every Banach space is reflexive. Bull. Am. Math. Soc. 67, 97–98 (1961)CrossRefGoogle Scholar
  8. 8.
    Bollobás, B.: An extension to the theorem of Bishop and Phelps. Bull. Lond. Math. Soc. 2, 181–182 (1970)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Bonsall, F.F., Duncan, J.: Numerical ranges of operators on normed spaces and of elements of normed algebras. In: London Mathematical Society Lecture Note Series, 2. Cambridge University Press, London, New York (1971)Google Scholar
  10. 10.
    Bonsall, F.F., Duncan, J.: Numerical ranges. II. In: London Mathematical Society Lecture Notes Series, No. 10. Cambridge University Press, New York, London (1973)Google Scholar
  11. 11.
    Bourgain, J.: On dentability and the Bishop–Phelps property. Israel J. Math. 78, 265–271 (1977)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Capel, A., Martín, M., Merí, J.: Numerical radius attaining compact linear operators. J. Math. Anal. Appl. 445, 1258–1266 (2017)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Chica, M., Kadets, V., Martín, M., Rambla-Barreno, F., Moreno-Pulido, S.: Bishop–Phelps–Bollobás modudi of a Banach space. J. Math. Anal. Appl. 412, 697–719 (2014)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Chica, M., Martín, M., Merí, J.: Numerical radius of rank-1 operators on Banach spaces. Quart. J. Math. 65, 89–100 (2014)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Cho, D.H., Choi, Y.S.: The Bishop–Phelps–Bollobás theorem on bounded closed convex sets. J. Lond. Math. Soc. 93, 502–518 (2016)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Dantas, S., García, D., Maestre, M., Martín, M.: The Bishop–Phelps–Bollobás property for compact operators. Can. J. Math. 70(1), 53–73 (2018)CrossRefGoogle Scholar
  17. 17.
    Dantas, S., Kadets, V., Kim, S. K., Lee, H. J., Martín, M.: On the pointwise Bishop–Phelps–Bollobás property for operators. Can. J. Math.  https://doi.org/10.4153/S0008414X18000032 (accepted)
  18. 18.
    Dantas, S., Kim, S.K., Lee, H.J.: The Bishop–Phelps–Bollobás point property. J. Math. Anal. Appl. 444, 1739–1751 (2016)MathSciNetCrossRefGoogle Scholar
  19. 19.
    García-Pacheco, F.J.: The AHSP is inherited by \(E\)-summands. Adv. Oper. Theory 2, 17–20 (2017)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Guirao, A., Kozhushkina, O.: The Bishop–Phelp–Bollobás property for numerical radius in \(\ell _1(\mathbb{C})\). Stud. Math. 218, 41–54 (2013)CrossRefGoogle Scholar
  21. 21.
    Hardtke, J.: Absolute sums of Banach spaces and some geometric properties related to rontundity and smoothness. Banach J. Math. Anal. 8, 295–334 (2014)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Harmand, P., Werner, D., Werner, D.: \(M\)-ideals in Banach Spaces and Banach Algebras. Lecture Notes in Mathematics, vol. 1547. Springer, Berlin (1993)CrossRefGoogle Scholar
  23. 23.
    Kim, S.K., Lee, H.J., Martín, M.: On the Bishop–Phelps–Bollobás property for numerical radius. Abstr. Appl. Anal. 2014, 15 (2014). Article ID 479208zbMATHGoogle Scholar
  24. 24.
    Kim, S.K., Lee, H.J., Martín, M., Merí, J.: On a second numerical index for Banach spaces. Proc. R. Soc. Edinb. Sect. A .  https://doi.org/10.1017/prm.2018.75 (accepted)
  25. 25.
    Kim, S.K., Lee, H.J., Martín, M.: The Bishop–Phelps–Bollobás theorem for operators from \(\ell _1\) sums. J. Math. Anal. Appl. 428, 920–929 (2015)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Kim, S.K., Lee, H.J., Martín, M.: On the Bishop–Phelps–Bollobás theorem for operators and numerical radius. Stud. Math. 233, 141–151 (2016)zbMATHGoogle Scholar
  27. 27.
    Lindenstrauss, J.: On operators which attain their norm. Israel J. Math. 1, 139–148 (1963)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Mena, J.F., Payá, R., Rodríguez, A.: Semisummands and semiideals in Banach spaces. Israel J. Math. 52, 33–67 (1985)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Mena, J.F., Payá, R., Rodríguez, A.: Absolute subspaces of Banach spaces. Quart. J. Math. 40, 33–37 (1989)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Mena, J.F., Payá, R., Rodríguez, A.: Absolutely proximinal subspaces of Banach spaces. J. Approx. Theory 65, 46–72 (1991)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Payá, R.: Técnicas de Rango Numérico y Estructura en Espacios Normados. Ph.D. Dissertation, Universidad de Granada, Spain (1980). Available at http://hdl.handle.net/10481/52674
  32. 32.
    Payá, R.: A counterexample on numerical radius attaining operators. Israel J. Math. 79, 83–101 (1992)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Payá, R., Saleh, Y.: Norm attaining operators from \(L_1(\mu )\) into \(L_{\infty }(\nu )\). Arch. Math. 75, 380–388 (2000)MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsPOSTECHPohangRepublic of Korea
  2. 2.Department of MathematicsFaculty of Electrical Engineering, Czech Technical University in PraguePrague 6Czech Republic
  3. 3.Departamento de Análisis Matemático, Facultad de CienciasUniversidad de GranadaGranadaSpain

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