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Matrix Domain of a Regular Matrix Derived by Euler Totient Function in the Spaces \(c_0\) and c

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Abstract

The main purpose of this manuscript is to introduce Banach spaces \(c_0^\Phi \) and \(c^\Phi \) as the matrix domain of a regular matrix \(\Phi \) derived by the Euler totient function. These spaces consist of \(\varphi \)-convergent to zero and \(\varphi \)-convergent sequences, respectively. After determining \(\alpha \)-, \(\beta \)- and \(\gamma \)-duals of these spaces, some matrix classes are characterized. Finally, using the Hausdorff measure of noncompactness, the characterization of some classes of compact operators on the space \(c_0^\Phi \) is given.

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References

  1. Mursaleen, M.: Generalized spaces of difference sequences. J. Math. Anal. Appl. 203(3), 738–745 (1996)

    Article  MathSciNet  Google Scholar 

  2. Mursaleen, M., Noman, A.K.: On the spaces of \(\lambda \)-convergent and bounded sequences. Thai J. Math. 8(2), 311–329 (2012)

    MathSciNet  MATH  Google Scholar 

  3. Başar, F., Kirişçi, M.: Almost convergence and generalized difference matrix. Comput. Math. Appl. 61, 602–611 (2011)

    Article  MathSciNet  Google Scholar 

  4. Candan, M.: Almost convergence and double sequential band matrix. Acta Math. Sci. Ser. B. Engl Edition 34B(2), 354–366 (2014)

    MathSciNet  MATH  Google Scholar 

  5. Candan, M.: Domain of the double sequential band matrix in the spaces of convergent and null sequences. Adv. Differ. Equ. 2014, 1–18 (2014)

    Article  MathSciNet  Google Scholar 

  6. Kara, E.E., İlkhan, M.: Some properties of generalized Fibonacci sequence spaces. Linear Multilinear Algebra 64(11), 2208–2223 (2016)

    Article  MathSciNet  Google Scholar 

  7. Ercan, S., Bektaş, Ç.A.: Some geometrical and topological properties of a new BK-space derived by using regular matrix of Fibonacci numbers. Linear Multilinear Algebra 65(5), 909–921 (2017)

    Article  MathSciNet  Google Scholar 

  8. Das, A., Hazarika, B.: Some new Fibonacci difference spaces of non-absolute type and compact operators. Linear Multilinear Algebra 65(12), 2551–2573 (2017)

    Article  MathSciNet  Google Scholar 

  9. Meng, J., Song, M.: On some binomial \(B^{(m)}\)-difference sequence spaces. J. Inequal. Appl. 2017, 194 (2017)

    Article  MathSciNet  Google Scholar 

  10. Demiriz, S., Duyar, O.: Domain of the generalized double Cesaro matrix in some paranormed spaces of double sequences. Tbilisi Mathe. J. 10(2), 43–56 (2017)

    Article  MathSciNet  Google Scholar 

  11. Duyar, O., Demiriz, S.: On vector-valued operator Riesz sequence spaces. Commun. Fac Sci. Univ. Ankara Ser. A1 68(1), 236–247 (2019)

    MathSciNet  Google Scholar 

  12. Dündar, E., Başar, F.: On the fine spectrum of the upper triangle double band matrix \(\Delta ^+\) on the sequence space \(c_0\). Mathe. Commun. 18, 337–348 (2013)

    MATH  Google Scholar 

  13. Sarıgöl, M.A.: Spaces of series summable by absolute Cesàro and matrix operators. Commun. Math. Appl. 7(1), 11–22 (2016)

    Google Scholar 

  14. Hazar, G.C., Sarıgöl, M.: On absolute Nörlund spaces and matrix operators. Acta Mathe. Sin. English Ser. 34(5), 812–826 (2018)

    Article  Google Scholar 

  15. Gökçe, F., Sarıgöl, M.A.: Generalization of the space \(\ell _(p)\) derived by absolute Euler summability and matrix operators. J. Inequal. Appl. 2018, 133 (2018)

    Article  MathSciNet  Google Scholar 

  16. Meng, J., Mei, L.: The matrix domain and the spectra of a generalized difference operator. J. Math. Anal. Appl. 470(2), 1095–1107 (2019)

    Article  MathSciNet  Google Scholar 

  17. Başar, F.: Summability Theory and Its Applications. Bentham Science Publishers, İstanbul (2012)

    Book  Google Scholar 

  18. Malkowsky, E., Rakocevic, V.: An introduction into the theory of sequence spaces and measure of noncompactness. Zbornik radova Matematicki Inst SANU Belgrade 9(17), 143–234 (2000)

    MathSciNet  MATH  Google Scholar 

  19. Mursaleen, M., Noman, A.K.: Compactness by the Hausdorff measure of noncompactness. Nonlinear Anal. 73(8), 2541–2557 (2010)

    Article  MathSciNet  Google Scholar 

  20. Mursaleen, M., Noman, A.K.: Applications of the Hausdorff measure of noncompactness in some sequence spaces of weighted means. Comput. Math. Appl. 60(5), 1245–1258 (2010)

    Article  MathSciNet  Google Scholar 

  21. Başar, F., Malkowsky, E.: The characterization of compact operators on spaces of strongly summable and bounded sequences. Appl. Math. Comput. 217, 5199–5207 (2011)

    MathSciNet  MATH  Google Scholar 

  22. Başarır, M., Kara, E.E.: On compact operators on the Riesz B(m)-difference sequence spaces. Iran. J. Sci. Technol. Trans. A. Sci. 35, 279–285 (2011)

    MathSciNet  MATH  Google Scholar 

  23. Kara, E.E., Başarır, M.: On some Euler \(B^{(m)}\) difference sequence spaces and compact operators. J. Math. Anal. Appl. 379, 499–511 (2011)

    Article  MathSciNet  Google Scholar 

  24. Başarır, M., Kara, E.E.: On some difference sequence spaces of weighted means and compact operators. Ann. Funct. Anal. 2, 114–129 (2011)

    Article  MathSciNet  Google Scholar 

  25. Başarır, M., Kara, E.E.: On the B-difference sequence space derived by generalized weighted mean and compact operators. J. Math. Anal. Appl. 391, 67–81 (2012)

    Article  MathSciNet  Google Scholar 

  26. Hazar, G.C., Sarıgöl, M.: Compact and matrix operators on the space \(\vert C,-1\vert _k\). J. Comput. Anal. Appl. 25(6), 1014–1024 (2018)

    MathSciNet  Google Scholar 

  27. Kovac, E.: On \(\varphi \) convergence and \(\varphi \) density. Math. Slovaca 55, 329–351 (2005)

    MathSciNet  MATH  Google Scholar 

  28. Niven, I., Zuckerman, H.S., Montgomery, H.L.: An Introduction to the Theory of Numbers, 5th edn. Wiley, New York (1991)

    MATH  Google Scholar 

  29. Schoenberg, I.: The integrability of certain functions and related summability methods. Am. Math. Mon. 66, 361–375 (1959)

    Article  MathSciNet  Google Scholar 

  30. İlkhan, M., Kara, E.E.: A new Banach space defined by Euler totient matrix operator. Oper. Matrices 13(2), 527–544 (2019)

    Article  MathSciNet  Google Scholar 

  31. Erdös, P.: Solutions of advanced problems: \(\phi \)-convergence. Am. Math. Mon. 85, 122–123 (1978)

    Article  Google Scholar 

  32. Mazur, S., Orlicz, W.: Sur les méthodes linéaires de sommation. C. R. Acad. Sei. Paris 196, 32–34 (1933)

    MATH  Google Scholar 

  33. Jarrah, A.M., Malkowsky, E.: Ordinary, absolute and strong summability and matrix transformations. Filomat. 17, 59–78 (2003)

    Article  MathSciNet  Google Scholar 

  34. Stieglitz, M., Tietz, H.: Matrix transformationen von folgenraumen eine ergebnisübersicht. Mathematische Zeitschrift 154, 1–16 (1977)

    Article  MathSciNet  Google Scholar 

  35. Altay, B., Başar, F.: Certain topological properties and duals of the matrix domain of a triangle matrix in a sequence space. J. Math. Anal. Appl. 336, 632–645 (2007)

    Article  MathSciNet  Google Scholar 

  36. Kirişçi, M., Başar, F.: Some new sequence spaces derived by the domain of generalized difference matrix. Comput. Math. Appl. 60, 1299–1309 (2010)

    Article  MathSciNet  Google Scholar 

  37. De Malafosse, B.: The Banach algebra \(\cal{B}(X)\), where \(X\) is a BK space and applications. Matematichki Vesnik 57(1–2), 41–60 (2005)

    MathSciNet  MATH  Google Scholar 

  38. Rakocevic, V.: Measures of noncompactness and some applications. Filomat 12(2), 87–120 (1998)

    MathSciNet  MATH  Google Scholar 

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  1. Department of Mathematics, Faculty of Science and Arts, Düzce University, Düzce, Turkey

    Merve İlkhan

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İlkhan, M. Matrix Domain of a Regular Matrix Derived by Euler Totient Function in the Spaces \(c_0\) and c. Mediterr. J. Math. 17, 27 (2020). https://doi.org/10.1007/s00009-019-1442-7

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  • DOI: https://doi.org/10.1007/s00009-019-1442-7

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