Almost convergence and Euler totient matrix

  • Serkan Demiriz
  • Merve İlkhan
  • Emrah Evren KaraEmail author
Original Paper


This paper is devoted to study the almost convergent sequence space \(\widehat{c}(\varPhi )\) derived by the Euler totient matrix. It is proved that the space \(\widehat{c}(\varPhi )\) and the space of all almost convergent sequences are linearly isomorphic. Further, the \(\beta \)-dual of the space \(\widehat{c}(\varPhi )\) is determined and Euler totient core of a complex-valued sequence has been defined. Finally, inclusion theorems related to this new type of core are obtained.


Euler function Almost convergence Euler totient matrix Möbius function Core theorems 

Mathematics Subject Classification

46A45 40A05 46A35 


  1. 1.
    Banach, S.: Théorie Des Opérations Linéaires. Chelsea Publishing Company, New York (1978)zbMATHGoogle Scholar
  2. 2.
    Başar, F.: \(f-\) conservative matrix sequences. Tamkang J. Math. 22, 205–212 (1991)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Başar, F., Çolak, R.: Almost-conservative matrix transformations. Turkish J. Math. 13, 91–100 (1989)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Başar, F., Kirişçi, M.: Almost convergence and generalized difference matrix. Comput. Math. Appl. 61, 602–611 (2011)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Başar, F., Solak, İ.: Almost-coercive matrix transformations. Rend. Mat. Appl. 11–2, 249–256 (1991)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Başarır, M., Kara, E.E.: On some difference sequence spaces of weighted means and compact operators. Ann. Funct. Anal. 2, 114–129 (2011)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Candan, M.: Vector-valued FK-spaces defined by a modulus function and an infinite matrix. Thai J. Math. 12(1), 155–165 (2014)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Cooke, R.G.: Infinite matrices and sequence spaces. MacMillan, New York (1950)zbMATHGoogle Scholar
  9. 9.
    Connor, J., Fridy, J.A., Orhan, C.: Core equality results for sequences. J. Math. Anal. Appl. 321, 515–523 (2006)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Çakan, C., Çoşkun, H.: Some new inequalities related to the invariant means and uniformly bounded function sequences. Appl. Math. Lett. 20, 605–609 (2007)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Çoşkun, H., Çakan, C.: A class of statistical and \(\sigma \)-conservative matrices. Czechoslovak Math. J. 55, 791–801 (2005)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Çoşkun, H., Çakan, C., Mursaleen, M.: On the statistical and \(\sigma \)-cores. Stud. Math. 154, 29–35 (2003)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Demirci, K.: \(A\)-statistical core of a sequence. Demonstratio Math. 33, 43–51 (2000)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Duran, J.P.: Infinite matrices and almost convergence. Math. Z. 128, 75–83 (1972)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Fridy, J.A., Orhan, C.: Statistical core theorems. J. Math. Anal. Appl. 208, 520–527 (1997)MathSciNetCrossRefGoogle Scholar
  16. 16.
    İlkhan, M., Kara, E.E.: A new Banach space defined by Euler totient matrix operator. Oper. Matrices 13(2), 527–544 (2019)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Karaisa, A., Karabıyık, Ü.: Almost sequence spaces derived by the domain of the matrix \(A^{r}\). Abstr. Appl. Anal. 2013, Art. ID 783731 (2013)Google Scholar
  18. 18.
    Kayaduman, K., Şengönül, M.: The spaces of Cesàro almost convergent sequences and core theorems. Acta Math. Sci. 32B, 2265–2278 (2012)CrossRefGoogle Scholar
  19. 19.
    Kılınç, G., Candan, M.: Some generalized Fibonacci difference spaces defined by a sequence of modulus functions. Facta Univ. Ser. Math. Inf. 32, 95–116 (2017)Google Scholar
  20. 20.
    Kirişçi, M.: The spaces of Euler almost null and Euler almost convergent sequences. Commun. Fac. Sci. Univ. Ank. Series A1(62), 85–100 (2013)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Kirişçi, M.: Almost convergence and generalized weighted mean II. J. Inequal. Appl. 2014, 93 (2014)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Lorentz, G.G.: A contribution to the theory of divergent sequences. Acta Math. 80, 167–190 (1948)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Kovac, E.: On \(\varphi \) convergence and \(\varphi \) density. Math. Slovaca 55, 329–351 (2005)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Mursaleen, M.: Invariant means and some matrix transformations. Indian J. Pure Appl. Math. 25, 353–359 (1994)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Mursaleen, M., Savaş, E., Aiyub, M., Mohuiddine, S.A.: Matrix transformations between the spaces of Cesàro sequences and invariant means. Int. J. Math. Math. Sci. 2006, Art. ID 74319 (2006)Google Scholar
  26. 26.
    Qamaruddin, Q., Mohuiddine, S.A.: Almost convergence and some matrix transformations. Filomat 21, 261–266 (2007)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Shcherbakov, A.A.: Kernels of sequences of complex numbers and their regular transformations. Math. Notes 22, 948–953 (1977)CrossRefGoogle Scholar
  28. 28.
    Sıddıqi, J.A.: Infinite matrices summing every almost periodic sequences. Pac. J. Math. 39, 235–251 (1971)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Simons, S.: Banach limits, infinite matrices and sublinear functionals. J. Math. Anal. Appl. 26, 640–655 (1969)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Sönmez, A.: Almost convergence and triple band matrix. Math. Comput. Model. 57, 2393–2402 (2012)CrossRefGoogle Scholar
  31. 31.
    Steinhaus, H.: Quality control by sampling. Collog. Math. 2, 98–108 (1951)CrossRefGoogle Scholar
  32. 32.
    Şengönül, M., Kayaduman, K.: On the Riesz almost convergent sequences space. Abstr. Appl. Anal. 2012, Art. ID 691694 (2012)Google Scholar
  33. 33.
    Schoenberg, I.: The integrability of certain functions and related summability methods. Am. Math. Mon. 66, 361–375 (1959)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Yeşilkayagil, M., Başar, F.: Spaces of \(A_{\lambda }-\)almost null and \(A_{\lambda }-\)almost convergence sequences. J. Egypt. Math. Soc. 23(1), 119–126 (2015)Google Scholar

Copyright information

© Tusi Mathematical Research Group (TMRG) 2019

Authors and Affiliations

  1. 1.Department of MathematicsTokat Gaziosmanpaşa UniversityTokatTurkey
  2. 2.Department of MathematicsDüzce UniversityDüzceTurkey

Personalised recommendations