On the Lipschitz numerical index of Banach spaces

• Choi, Geunsu ^[1] ; Jung, Mingu ^[2] ; Tag, Hyung-Joon ^[3]
  1. [1] Sunchon National University

     Sunchon National University

     Corea del Sur

     
  2. [2] Korea Institute for Advanced Study

     Korea Institute for Advanced Study

     Corea del Sur

     
  3. [3] Sejong University

     Sejong University

     Corea del Sur

     

  Mostrar afiliaciones +
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 76, Fasc. 1, 2025, págs. 81-103
• Idioma: inglés
• DOI: 10.1007/s13348-023-00421-9
• Texto completo no disponible (Saber más ...)
• Resumen

  • In this article, we investigate further on the Lipschitz numerical radius and index which were recently introduced. First, we provide some renorming results on Lipschitz numerical index and introduce a concept of Lipschitz numerical radius attaining maps. Namely, we observe that for any Banach space X, the set of Lipschitz numerical indices of Banach spaces which are isomorphic to X is an interval. Moreover, we show the set of Lipschitz numerical radius attaining maps is not dense in the space of Lipschitz maps vanishing at zero. Next, we discuss the Lipschitz numerical index of vector-valued function spaces, absolute sums of Banach spaces, the Köthe–Bochner spaces, and Banach spaces which contain a dense union of increasing family of one-complemented subspaces.

• Referencias bibliográficas
  • Acosta, M., Kim, S.G.: Denseness of holomorphic functions attaining their numerical radii. Isr. J. Math. 161, 373–386 (2007)
  • Acosta, M.D., Aguirre, F.J., Payá, R.: A space by W. Gowers and new results on norm and numerical radius attaining operators. Acta Univ. Carolin....
  • Benyamini, Y., Lindenstrauss, J.: Geometric Nonlinear Functional Analysis, vol.1. American Mathematical Society (2000)
  • Bonsall, F.F., Duncan, J.: Numerical Ranges II, London Mathematical Society. Lecture Note Series 10. Cambridge University Press (1973)
  • Boyko, K., Kadets, V., Martín, M., Werner, D.: Numerical index of Banach spaces and duality. Math. Proc. Camb. Philos. Soc. 142, 93–102 (2007)
  • Capel, A., Martín, M., Merí, J.: Numerical radius attaining compact linear operators. J. Math. Anal. Appl. 445, 1258–1266 (2017)
  • Cardassi, C.S.: Numerical radius-attaining operators on C(K). Proc. Am. Math. Soc. 95, 537–543 (1985)
  • Cascales, B., Guirao, A., Kadets, V.: A Bishop–Phelps–Bollobás type theorem for uniform algebras. Adv. Math. 240, 370–382 (2013)
  • Choi, Y.S., Kim, S.G.: Norm or numerical radius attaining multilinear mappings and polynomials. J. Lond. Math. Soc. 54, 135–147 (1996)
  • Choi, Y.S., García, D., Kim, S.G., Maestre, M.: The polynomial numerical index of a Banach space. Proc. Edin. Math. Soc. 49, 39–52 (2006)
  • Choi, Y.S., García, D., Maestre, M., Martín, M.: Polynomial numerical index for some complex vector-valued function spaces. Q. J. Math. 59,...
  • Dales, H.G.: Banach Algebras and Automatic Continuity, vol. 24, London Mathematical Society Monographs. New Series, The Clarendon Press, Oxford...
  • Daugavet, I.K.: A property of completely continuous operators in the space . Uspekhi Mat. Nauk. 18(5), 157–158 (1963).
  • Duncan, J., McGregor, C.M., Pryce, J.D., White, A.J.: The numerical index of a normed space. J. Lond. Math. Soc. 2, 481–488 (1970)
  • Falcó, J., García, D., Kim, S.K., Lee, H.J., Maestre, M.: Polarization constant for the numerical radius. Mediterr. J. Math. 17, 1–12 (2020)
  • Finet, C., Martín, M., Paýa, R.: Numerical index and renorming. Proc. Am. Math. Soc. 131, 871–877 (2003)
  • García, D., Grecu, B.C., Maestre, M., Martín, M., Merí, J.: Polynomial numerical indices of C(K) and L1(u). Proc. Am. Math. Soc. 142(4), 1229–1235...
  • Harris, L.A.: The numerical range of holomorphic functions in Banach spaces. Am. J. Math. 93, 1005–1019 (1971)
  • Kadets, V., Martín, M., Payá, R.: Recent progress and open questions on the numerical index of Banach spaces. Rev. R. Acad. Cien. Serie A....
  • Kadets, V., Martín, M., Merí, J., Werner, D.: Lipschitz slices and the Daugavet equation for Lipschitz operators. Proc. Am. Math. Soc. 143,...
  • Kadets, V., Martín, M., Soloviova, M.: Norm-attaining Lipschitz functionals. Banach J. Math. Anal. 10, 621–637 (2016)
  • Kim, S.K., Lee, H.J.: A Urysohn-type theorem and the Bishop–Phelps–Bollobás theorem for holomorphic Functions. J. Math. Anal. Appl. 480, 123393...
  • Lee, H.J., Tag, H.J.: Diameter two properties in some vector-valued function spaces. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A. Mat. (RACSAM)....
  • Lee, H.J.: Generalized numerical index of function algebras. J. Funct. Spaces 9080867, 1–6 (2019)
  • Lin, P.K.: Köthe–Bochner function spaces. Birkäuser, Basel (2012)
  • Lindenstrauss, J., Tzafriri, L.: Classical Banach spaces II. Springer, Berlin (1979)
  • López, G., Martín, M., Merí, J.: Numerical index of Banach spaces of weakly or weakly-star continuous functions. Rocky Mt. J. Math. 38, 213–223...
  • Lozanovskii, G. Ya.: On almost integral operators in KB-spaces. Vestnik Leningr. Univ. Mat. Mekh. Astr 21, 35–44 (1966)
  • Martín, M., Payá, R.: Numerical index of vector-valued function spaces. Studia Math. 142(3), 269–280 (2000)
  • Martín, M., Merí, J., Popov, M., Randrianantoanina, B.: Numerical index of absolute sums of Banach spaces. J. Math. Anal. and Appl 375, 207–222...
  • McGuigan, R.A.: On the connectedness of isomorphism classes. Manuscr. Math. 3, 1–5 (1970)
  • Payá, R.: A counterexample on numerical radius attaining operators. Isr. J. Math. 79, 83–101 (1992)
  • Rudin, W.: Real and Complex Analysis. McGraw-Hill, New York (1987)
  • Sims, B.: On numerical range and its application to Banach algebras. Ph.D. Dissertation, University of Newcastle, Australia (1972)
  • Wang, R.: The numerical index of Lipschitz operators on Banach spaces. Studia Math. 209(1), 43–51 (2012)
  • Wang, R., Huang, X., Tan, D.: On the numerical radius of Lipschitz operators in Banach spaces. J. Math. Anal. Appl. 411, 1–18 (2014)
  • Werner, D.: Recent progress on the Daugavet property. Ir. Math. Soc. Bull. 46, 77–97 (2001)
  • Wojtaszczyk, P.: Some remarks on the Daugavet equation. Proc. Am. Math. Soc. 115, 1047–1052 (1992)
  • Zarantonello, E.H.: The closure of the numerical range contains the spectrum. Pac. J. Math. 22, 575–595 (1967)