Historical Introduction: A Walk on the Results for Banach Spaces with Numerical Index 1

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


This chapter contains an overview of the known results about Banach spaces with numerical index 1, as well as the notation and terminology we will need along the book.


  1. 1.
    Y. Abramovich, A generalization of a theorem of J. Holub. Proc. Am. Math. Soc. 108, 937–939 (1990)Google Scholar
  2. 2.
    M. Acosta, CL-spaces and numerical radius attaining operators. Extracta Math. 5, 138–140 (1990)Google Scholar
  3. 3.
    M. Acosta, Operadores que alcanzan su radio numérico. Tesis doctoral, Secretariado de publicaciones, Universidad de Granada, 1990Google Scholar
  4. 5.
    M. Acosta, A. Kaminska, M. Mastylo, The Daugavet property in rearrangement invariant spaces. Trans. Am. Math. Soc. 367, 4061–4078 (2015)Google Scholar
  5. 6.
    E. Alfsen, E. Effros, Structure in real Banach spaces. Ann. Math. 96, 98–173 (1972)Google Scholar
  6. 7.
    M. Ardalani, Numerical index with respect to an operator. Stud. Math. 224, 165–171 (2014)Google Scholar
  7. 9.
    A. Avilés, V. Kadets, M. Martín, J. Merí, V. Shepelska, Slicely countably determined Banach spaces. Trans. Am. Math. Soc. 362, 4871–4900 (2010)Google Scholar
  8. 11.
    Y. Benyamini, J. Lindenstrauss, Geometric Nonlinear Functional Analysis, Vol. 1. American Mathematical Society Colloquium Publications, vol. 48 (AMS, Providence, RI, 2000)Google Scholar
  9. 12.
    D. Bilik, V. Kadets, R. Shvidkoy, G. Sirotkin, D. Werner, Narrow operators on vector-valued sup-normed spaces. Ill. J. Math. 46(2), 421–441 (2002)Google Scholar
  10. 13.
    H. Bohnenblust, S. Karlin, Geometrical properties of the unit sphere in Banach algebras. Ann. Math. 62, 217–229 (1955)Google Scholar
  11. 14.
    B. Bollobás, An extension to the Theorem of Bishop and Phelps. Bull. Lond. Math. Soc. 2, 181–182 (1970)Google Scholar
  12. 15.
    F. Bonsall, J. Duncan, Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras. London Mathematical Society. Lecture Note Series, vol. 2 (Cambridge University Press, Cambridge, 1971)Google Scholar
  13. 16.
    F. Bonsall, J. Duncan, Numerical Ranges II. London Mathematical Society. Lecture Note Series, vol. 10 (Cambridge University Press, Cambridge, 1973)Google Scholar
  14. 17.
    T. Bosenko, Strong Daugavet operators and narrow operators with respect to Daugavet centers. Visn. Khark. Univ. Ser. Mat. Prykl. Mat. Mekh. 931(62), 5–19 (2010)Google Scholar
  15. 18.
    T. Bosenko, V. Kadets, Daugavet centers. Zh. Mat. Fiz. Anal. Geom. 6, 3–20 (2010)Google Scholar
  16. 19.
    G. Botelho, E. Santos, Representable spaces have the polynomial Daugavet property. Arch. Math. 107, 37–42 (2016)Google Scholar
  17. 20.
    R. Bourgin, Geometric Aspects of Convex Sets with the Radon-Nikodym Property. Lecture Notes in Mathematics, vol. 993 (Springer, Berlin, 1983)Google Scholar
  18. 21.
    K. Boyko, V. Kadets, M. Martín, D. Werner, Numerical index of Banach spaces and duality. Math. Proc. Camb. 142, 93–102 (2007)Google Scholar
  19. 22.
    K. Boyko, V. Kadets, M. Martín, J. Merí, Properties of lush spaces and applications to Banach spaces with numerical index 1. Stud. Math. 190, 117–133 (2009)Google Scholar
  20. 23.
    M. Cabrera, A. Rodríguez-Palacios, Non-associative normed algebras, Volume 1: The Vidav-Palmer and Gelfand-Naimark theorems, in Encyclopedia of Mathematics and Its Applications, vol. 154 (Cambridge University Press, Cambridge, 2014)Google Scholar
  21. 27.
    Y. Choi, S. Kim, The Bishop-Phelps-Bollobás property and lush spaces. J. Math. Anal. Appl. 390, 549–555 (2012)Google Scholar
  22. 28.
    Y. Choi, D. García, M. Maestre, M. Martín, The Daugavet equation for polynomials. Stud. Math. 178, 63–82 (2007)Google Scholar
  23. 30.
    M. Crabb, J. Duncan, C. McGregor, Mapping theorems and the numerical radius. Proc. Lond. Math. Soc. 25, 486–502 (1972)Google Scholar
  24. 32.
    E. Dancer, B. Sims, Weak star separability. Bull. Aust. Math. Soc. 20, 253–257 (1979)Google Scholar
  25. 33.
    I. Daugavet, On a property of completely continuous operators in the space C. Usp. Mat. Nauk 18(5), 157–158 (1963) (Russian)Google Scholar
  26. 36.
    J. Diestel, J. Uhl, Vector Measures. Mathematical Surveys, No. 15 (AMS, Providence, RI, 1977), XIIIGoogle Scholar
  27. 37.
    J. Duncan, C. McGregor, J. Pryce, A. White, The numerical index of a normed space. J. Lond. Math. Soc. 2, 481–488 (1970)Google Scholar
  28. 39.
    C. Foiaş, I. Singer, Points of diffusion of linear operators and almost diffuse operators in spaces of continuous functions. Math. Z. 87, 434–450 (1965)Google Scholar
  29. 40.
    V. Fonf, One property of Lindenstrauss-Phelps spaces. Funct. Anal. Appl. 13, 66–67 (1979)Google Scholar
  30. 41.
    R. Fullerton, Geometrical characterization of certain function spaces, in Proceedings of the International Symposium Linear Spaces (Jerusalem 1960) (Jerusalem Academic Press/Pergamon, Jerusalem/Oxford, 1961), pp. 227–236Google Scholar
  31. 43.
    D. García, B. Grecu, M. Maestre, M. Martín, J. Merí, Polynomial numerical indices of C(K) and L 1(μ). Proc. Am. Math. Soc. 142, 1229–1235 (2014)Google Scholar
  32. 44.
    N. Ghoussoub, G. Godefroy, B. Maurey, W. Schachermayer, Some Topological and Geometrical Structures in Banach Spaces. Memoirs of the AMS (AMS, Providence, RI, 1987)Google Scholar
  33. 45.
    B. Glickfeld, On an inequality of Banach algebra geometry and semi-inner-product space theory. Ill. J. Math. 14, 76–81 (1970)Google Scholar
  34. 49.
    K. Gustafson, D. Rao, Numerical Range. The Field of Values of Linear Operators and Matrices. Universitext (Springer, New York, 1997)Google Scholar
  35. 50.
    P. Halmos, A Hilbert Space Problem Book (Van Nostrand, New York, 1967)Google Scholar
  36. 51.
    O. Hanner, Intersections of translates of convex bodies. Math. Scan. 4, 65–87 (1956)Google Scholar
  37. 52.
    A. Hansen, Å. Lima, The structure of finite dimensional Banach spaces with the 3.2. intersection properties. Acta Math. 146, 1–23 (1981)Google Scholar
  38. 55.
    J. Holub, A property of weakly compact operators on C[0, 1]. Proc. Am. Math. Soc. 97, 396–398 (1986)Google Scholar
  39. 58.
    V. Kadets, Some remarks concerning the Daugavet equation. Quaest. Math. 19, 225–235 (1996)Google Scholar
  40. 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
  41. 61.
    V. Kadets, R. Shvidkoy, G. Sirotkin, D. Werner, Banach spaces with the Daugavet property. Trans. Am. Math. Soc. 352, 855–873 (2000)Google Scholar
  42. 62.
    V. Kadets, R. Shvidkoy, D. Werner, Narrow operators and rich subspaces of Banach spaces with the Daugavet property. Stud. Math. 147, 269–298 (2001)Google Scholar
  43. 64.
    V. Kadets, M. Martín, R. Payá, Recent progress and open questions on the numerical index of Banach spaces. Rev. R. Acad. Cienc. Exactas Fís. Natl. Ser. A Math. 100, 155–182 (2006)Google Scholar
  44. 65.
    V. Kadets, M. Martín, J. Merí, Norm equalities for operators on Banach spaces. Indiana U. Math. J. 56, 2385–2411 (2007)Google Scholar
  45. 66.
    V. Kadets. M. Martín, J. Merí, R. Payá, Convexity and smoothness of Banach spaces with numerical index one. Ill. J. Math. 53, 163–182 (2009)Google Scholar
  46. 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
  47. 68.
    V. Kadets, M. Martín, J. Merí, D. Werner, Lushness, numerical index 1 and the Daugavet property in rearrangement invariant spaces. Can. J. Math. 65, 331–348 (2013)MathSciNetCrossRefGoogle Scholar
  48. 69.
    V. Kadets, M. Martín, J. Merí, D. Werner, Lipschitz slices and the Daugavet equation for Lipschitz operators. Proc. Am. Math. Soc. 143, 5281–5292 (2015)MathSciNetCrossRefGoogle Scholar
  49. 70.
    V. Kadets, A. Pérez, D. Werner, Operations with slicely countably determined sets. Funct. Approx. (to appear). arXiv:1708.05218
  50. 71.
    V. Kadets, M. Martín, G. López, D. Werner, Equivalent norms with an extremely nonlineable set of norm attaining functionals. J. Inst. Math. Jussieu (to appear). arXiv:1709.01756,
  51. 72.
    A. Kaidi, A. Morales, A. Rodríguez-Palacios, Non associative C -algebras revisited, in Recent Progress in Functional Analysis, ed. by K.D. Bierstedt, J. Bonet, M. Maestre, J. Schmets. Proceedings of the International Function Analysis Meeting on the Occasion of the 70th Birthday of Professor Manuel Valdivia (Elsevier, Amsterdam, 2001), pp. 379–408Google Scholar
  52. 73.
    S. Kim, M. Martín, J. Merí, On the polynomial numerical index of the real spaces c 0, 1 and . J. Math. Anal. Appl. 337, 98–106 (2008)MathSciNetCrossRefGoogle Scholar
  53. 74.
    S. Kim, H. Lee, M. Martín, On the Bishop-Phelps-Bollobás theorem for operators and numerical radius. Stud. Math. 233, 141–151 (2016)zbMATHGoogle Scholar
  54. 75.
    P. Koszmider, M. Martín, J. Merí, Isometries on extremely non-complex C(K) spaces. J. Inst. Math. Jussieu 10, 325–348 (2011)MathSciNetCrossRefGoogle Scholar
  55. 76.
    K. Kunen, H. Rosenthal, Martingale proofs of some geometrical results in Banach space theory. Pac. J. Math. 100, 153–175 (1982)MathSciNetCrossRefGoogle Scholar
  56. 77.
    H. Lee, M. Martín, Polynomial numerical indices of Banach spaces with 1-unconditional bases. Linear Algebra Appl. 437, 2011–2008 (2012)MathSciNetzbMATHGoogle Scholar
  57. 78.
    Å. Lima, Intersection properties of balls and subspaces in Banach spaces. Trans. Am. Math. Soc. 227, 1–62 (1977)MathSciNetCrossRefGoogle Scholar
  58. 79.
    Å. Lima, Intersection properties of balls in spaces of compact operators. Ann. Inst. Fourier (Grenoble) 28, 35–65 (1978)MathSciNetCrossRefGoogle Scholar
  59. 80.
    J. Lindenstrauss, Extension of Compact Operators. Memoirs of the American Mathematical Society, vol. 48 (American Mathematical Society, Providence, RI, 1964)Google Scholar
  60. 81.
    J. Lindenstrauss, R. Phelps, Extreme point properties of convex bodies in reflexive Banach spaces. Isr. J. Math. 6, 39–48 (1968)MathSciNetCrossRefGoogle Scholar
  61. 82.
    J. Lindenstrauss, L. Tzafriri, Classical Banach Spaces I: Sequence Spaces (Springer, Berlin, 1977)CrossRefGoogle Scholar
  62. 83.
    G. López, M. Martín, R. Payá, Real Banach spaces with numerical index 1. Bull. Lond. Math. Soc. 31, 207–212 (1999)MathSciNetCrossRefGoogle Scholar
  63. 84.
    G. Lozanovskii, On almost integral operators in KB-spaces. Vestnik Leningrad Univ. Mat. Mekh. Astr. 21(7), 35–44 (1966) (Russian)MathSciNetGoogle Scholar
  64. 86.
    G. Lumer, Semi-inner-product spaces. Trans. Am. Math. Soc. 100, 29–43 (1961)MathSciNetCrossRefGoogle Scholar
  65. 87.
    M. Martín, A survey on the numerical index of a Banach space. Extracta Math. 15(2), 265–276 (2000)MathSciNetzbMATHGoogle Scholar
  66. 88.
    M. Martín, Banach spaces having the Radon-Nikodým property and numerical index 1. Proc. Am. Math. Soc. 131, 3407–3410 (2003)CrossRefGoogle Scholar
  67. 89.
    M. Martín, The group of isometries of a Banach space and duality. J. Funct. Anal. 255, 2966–2976 (2008)MathSciNetCrossRefGoogle Scholar
  68. 90.
    M. Martín, The alternative Daugavet property for C -algebras and JB -triples. Math. Nachr. 281, 376–385 (2008)MathSciNetCrossRefGoogle Scholar
  69. 91.
    M. Martín, Positive and negative results on the numerical index of Banach spaces and duality. Proc. Am. Math. Soc. 137, 3067–3075 (2009)MathSciNetCrossRefGoogle Scholar
  70. 92.
    M. Martín, On different definitions of numerical range. J. Math. Anal. Appl. 433, 877–886 (2016)MathSciNetCrossRefGoogle Scholar
  71. 93.
    M. Martín, J. Merí, Numerical index of some polyhedral norms on the plane. Linear Multilinear Algebra 55, 175–190 (2007)MathSciNetCrossRefGoogle Scholar
  72. 94.
    M. Martín, T. Oikhberg, An alternative Daugavet property. J. Math. Anal. Appl. 294, 158–180 (2004)MathSciNetCrossRefGoogle Scholar
  73. 95.
    M. Martín, R. Payá, Numerical index of vector-valued function spaces. Stud. Math. 142, 269–280 (2000)MathSciNetCrossRefGoogle Scholar
  74. 96.
    M. Martín, R. Payá, On CL-spaces and almost-CL-spaces. Ark. Mat. 42, 107–118 (2004)MathSciNetCrossRefGoogle Scholar
  75. 97.
    M. Martín, A. Villena, Numerical index and Daugavet property for L (μ, X). Proc. Edinb. Math. Soc. 46, 415–420 (2003)Google Scholar
  76. 98.
    M. Martín, J. Merí, R. Payá, On the intrinsic and the spatial numerical range. J. Math. Anal. Appl. 318, 175–189 (2006)MathSciNetCrossRefGoogle Scholar
  77. 99.
    M. Martín, J. Merí, M. Popov, On the numerical index of real L p(μ)-spaces. Isr. J. Math. 184, 183–192 (2011)MathSciNetCrossRefGoogle Scholar
  78. 100.
    J. Martínez, J.F. Mena, R. Payá, A. Rodríguez-Palacios, An approach to numerical ranges without Banach algebra theory. Ill. J. Math. 29(4), 609–626 (1985)MathSciNetzbMATHGoogle Scholar
  79. 101.
    C. McGregor, Finite dimensional normed linear spaces with numerical index 1. J. Lond. Math. Soc. 3, 717–721 (1971)MathSciNetCrossRefGoogle Scholar
  80. 102.
    R. Phelps, Convex Functions, Monotone Operators and Differentiability. Lecture Notes in Mathematics, vol. 1364 (Springer, Berlin, 1993)Google Scholar
  81. 103.
    E. Pipping, L- and M-structure in lush spaces, Zh. Mat. Fiz. Anal. Geom. 7, 87–95 (2011)MathSciNetzbMATHGoogle Scholar
  82. 104.
    A. Plichko, M. Popov, Symmetric function spaces on atomless probability spaces. Dissert. Math. 306, 1–85 (1990)MathSciNetzbMATHGoogle Scholar
  83. 105.
    M. Popov, B. Randrianantoanina, Narrow Operators on Function Spaces and Vector Lattices. De Gruyter Studies in Mathematics, vol. 45 (de Gruyter, Berlin, 2012)Google Scholar
  84. 106.
    S. Reisner, Certain Banach spaces associated with graphs and CL-spaces with 1-unconditional bases. J. Lond. Math. Soc. 43, 137–148 (1991)MathSciNetCrossRefGoogle Scholar
  85. 107.
    A. Rodríguez-Palacios, Banach space characterizations of unitaries: a survey. J. Math. Anal. Appl. 369, 168–178 (2010)MathSciNetCrossRefGoogle Scholar
  86. 108.
    H. Rosenthal, A characterization of Banach spaces containing l 1. Proc. Natl. Acad. Sci. U.S.A. 71, 2411–2413 (1974)MathSciNetCrossRefGoogle Scholar
  87. 109.
    H. Rosenthal, The Lie algebra of a Banach space, in Banach Spaces (Columbia, MO, 1984). Lecture Notes in Mathematics, vol. 1166 (Springer, Berlin, 1985), pp. 129–157Google Scholar
  88. 111.
    E. Sánchez-Pérez, D. Werner, Slice continuity for operators and the Daugavet property for bilinear maps. Funct. Approx. Comment. Math. 50, 251–269 (2014)MathSciNetzbMATHGoogle Scholar
  89. 112.
    K. Schmidt, Daugavet’s equation and orthomorphisms. Proc. Am. Math. Soc. 108, 905–911 (1990)MathSciNetzbMATHGoogle Scholar
  90. 114.
    R. Shvidkoy, Geometric aspects of the Daugavet property. J. Funct. Anal. 176, 198–212 (2000)MathSciNetCrossRefGoogle Scholar
  91. 116.
    D. Tan, X. Huang, R. Liu, Generalized-lush spaces and the Mazur-Ulam property. Stud. Math. 219, 139–153 (2013)MathSciNetCrossRefGoogle Scholar
  92. 117.
    O. Toeplitz, Das algebraische Analogon zu einem Satze von Fejer. Math. Z. 2, 187–197 (1918)MathSciNetCrossRefGoogle Scholar
  93. 118.
    D. Van Duslt, Characterizations of Banach Spaces Not Containing 1. CWI Tract, vol. 59 (Stichting Mathematisch Centrum/Centrum voor Wiskunde en Informatica, Amsterdam, 1989)Google Scholar
  94. 122.
    L. Weis, D. Werner, The Daugavet equation for operators not fixing a copy of C[0, 1]. J. Oper. Theory 39, 89–98 (1998)Google Scholar
  95. 123.
    D. Werner, The Daugavet equation for operators on function spaces. J. Funct. Anal. 143, 117–128 (1997)MathSciNetCrossRefGoogle Scholar
  96. 124.
    D. Werner, Recent progress on the Daugavet property. Irish. Math. Soc. Bull. 46, 77–97 (2001)MathSciNetzbMATHGoogle Scholar
  97. 125.
    P. Wojtaszczyk, Some remarks on the Daugavet equation. Proc. Am. Math. Soc. 115, 1047–1052 (1992)MathSciNetCrossRefGoogle Scholar

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© 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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