Power Series of Positive Linear Operators

  • Tuncer AcarEmail author
  • Ali Aral
  • Ioan Raşa


We describe a unifying approach for studying the power series of the positive linear operators from a certain class. For the same operators, we give simpler proofs of some known ergodic theorems.


Power series Ergodic theorem positive linear operator Eigenstructure \(C_{0}\)-semigroup 

Mathematics Subject Classification

Primary 41A36 47D06 37A30 15A18 



  1. 1.
    Abel, U.: Geometric series of Bernstein–Durrmeyer operators. East J. Approx. 15, 439–450 (2009)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Abel, U., Ivan, M., Paltanea, R.: Geometric series of Bernstein operators revisited. J. Math. Anal. Appl. 400, 22–24 (2013)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Abel, U., Ivan, M., Paltanea, R.: Geometric series of positive linear operators and the inverse Voronovskaya theorem on a compact interval. J. Approx. Theory 184, 163–175 (2014)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Acar, T., Aral, A., Rasa, I.: Power series of beta operators. Appl. Math. Comput. 247, 815–823 (2014)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Altomare, F., Campiti, M.: Korovkin Type Approximation Theory and Its Applications. W. de Gruyter, Berlin (1994)CrossRefGoogle Scholar
  6. 6.
    Altomare, F., Cappelletti Montano, M., Leonessa, V., Rasa, I.: Markov operators, positive semigroups and approximation processes. In: De Gruyter studies in mathematics, vol. 61. Walter de Gruyter GmbH, Berlin, Boston (2014)Google Scholar
  7. 7.
    Altomare, F., Rasa, I.: Lipschitz contractions, unique ergodicity and asymptotics of Markov semigroups. Bollettino UMI 9(5), 1–17 (2012)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Attalienti, A., Rasa, I.: The eigenstructure of some positive linear operators. Anal. Numer. Theor. Approx. 43, 45–58 (2014)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Cooper, S., Waldron, S.: The eigenstructure of Bernstein operators. J. Approx. Theory 105, 133–165 (2000)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Goldstein, J.A.: Semigroups of Linear Operators and Applications. Oxford Univ. Press, New York (1985)zbMATHGoogle Scholar
  11. 11.
    Gonska, H., Rasa, I., Stanila, E.D.: Power series of operators \(\ U_{n}^{\rho }\). Positivity 19, 237–249 (2015)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Heilmann, M., Rasa, I.: Eigenstructure and iterates for uniquely ergodic Kantorovich modifications of operators. Positivity 21, 897–910 (2017)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Heilmann, M., Rasa, I., \(C_{0}\)-semigroups associated with uniquely ergodic Kantorovich modifications of operators, Positivity. MathSciNetCrossRefGoogle Scholar
  14. 14.
    Nagel, R. (Ed.), One parameter semigroups of positive operators, Lecture Notes Math., vol 1184, Springer-Verlag, (1986)Google Scholar
  15. 15.
    Paltanea, R.: The power series of Bernstein operators. Autom. Comput. Appl. Math. 15, 247–253 (2006)MathSciNetGoogle Scholar
  16. 16.
    Pazy, A.: Semigroups of linear operators and applications to partial differential equations. Springer, New York (1983)CrossRefGoogle Scholar
  17. 17.
    Rasa, I.: Positive operators, Feller semigroups and diffusion equations associated with Altomare projections. Conf. Sem. Math. Univ. Bari 284, 1–26 (2002)MathSciNetGoogle Scholar
  18. 18.
    Rasa, I.: Power series of Bernstein operators and approximation of resolvents. Mediterr. J. Math. 9, 635–644 (2012)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Vladislav, T., Rasa, I.: Analiză Numerică. In: Aproximare, problema lui Cauchy abstractă, proiectori Altomare, p. 173. Editura Tehnică, Bucureşti (1999). ISBN 973-31-1336-0. (in Romanian)Google Scholar

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Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceSelcuk UniversityKonyaTurkey
  2. 2.Department of MathematicsKirikkale UniversityKirikkaleTurkey
  3. 3.Department of MathematicsTechnical University of Cluj-NapocaCluj-NapocaRomania

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