Korovkin-Type Results on Convergence of Sequences of Positive Linear Maps on Function Spaces

  • Maliheh Hosseini
  • Juan J. FontEmail author


In this paper, we deal with the convergence of sequences of positive linear maps to a (not assumed to be linear) isometry on spaces of continuous functions. We obtain generalizations of known Korovkin-type results and provide several illustrative examples.


Function space Korovkin’s theorem Choquet boundary Positive linear map 

Mathematics Subject Classification

Primary 41A36 Secondary 46E15 



  1. 1.
    Altomare, F.: Korovkin-type theorems and approximation by positive linear operators. Surv. Approx. Theory 5, 92–164 (2010)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Altomare, F., Campiti, M.: Korovkin-Type Approximation Theory and its Applications, de Gruyter Studies in Mathematics, 17. Walter de Gruyter Co., Berlin (1994)CrossRefGoogle Scholar
  3. 3.
    Araujo, J., Font, J.J.: Linear isometries between subspaces of continuous functions. Trans. Am. Math. Soc. 349, 413–428 (1997)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bishop, E., de Leeuw, K.: The representation of linear functionals by measures on sets of extreme points. Ann. Inst. Fourier (Grenoble) 9, 305–331 (1959)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Browder, A.: Introduction to Function Algebras. W. A. Benjamin, New York-Amsterdam (1969)zbMATHGoogle Scholar
  6. 6.
    Donner, K.: Extension of Positive Operators and Korovkin Theorems, Lecture Notes in Math. 904, Springer, Berlin, (1982)Google Scholar
  7. 7.
    Hachiro, T., Okayasu, T.: Some theorems of Korovkin type. Studia Math. 155(2), 131–143 (2003)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Korovkin, P.P.: On convergence of linear positive operators in the space of continuous functions. Doklady Akad. Nauk SSSR (N.S.) 90, 961–964 (1953). (in Russian)MathSciNetGoogle Scholar
  9. 9.
    Koshimizu, H., Miura, T., Takagi, H., Takahasi, S.E.: Real-linear isometries between subspaces of continuous functions. J. Math. Anal. Appl. 413, 229–241 (2014)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Mazur, S., Ulam, S.: Sur les transformations isométriques d’espaces vectoriels normés. C. R. Math. Acad. Sci. Paris 194, 946–948 (1932)zbMATHGoogle Scholar
  11. 11.
    Morozov, E.N.: Convergence of a sequence of positive linear operators in the space of continuous 2\(\pi \)-periodic functions of two variables. Kalinin. Gos. Ped. Inst. Uchen. Zap. 26, 129–142 (1958). (in Russian)MathSciNetGoogle Scholar
  12. 12.
    Phelps, R.R.: Lectures on Choquet’s Theorem, 2nd ed., Lecture Notes in Math. 1757, Springer, Berlin, (2001)Google Scholar
  13. 13.
    Phelps, R.R.: The range of Tf for certain linear operators \(T\). Proc. Am. Math. Soc. 16, 381–382 (1965)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Scheffold, E.: Uber die punktweise konvergenz von operatoren in \(C(X)\). Rev. Acad. Ci. Zaragoza 28, 5–12 (1973)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Volkov, V.I.: Conditions for convergence of a sequence of positive linear operators in the space of continuous functions of two variables. Kalinin. Gos. Ped. Inst. Uchen. Zap. 26, 27–40 (1958). (in Russian)MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsK. N. Toosi University of TechnologyTehranIran
  2. 2.Departamento de MatemáticasUniversitat Jaume ICastellónSpain

Personalised recommendations