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Some Results Related to New Jordan Totient Double Sequence Spaces

Year 2022, Volume: 14 Issue: 2, 271 - 280, 30.12.2022
https://doi.org/10.47000/tjmcs.1007927

Abstract

The 4 dimensional (4d) Jordan totient matrix which is described by the aid of the famous Jordan's function and some new Jordan totient double sequence spaces described as the domain of this aforementioned matrix have been examined by Erdem and Demiriz . In the present paper, first of all we define two new double sequence spaces by using the 4d Jordan totient matrix and we show that this newly described double sequence spaces are Banach spaces with their norm. Then, we give some inclusion relations including this spaces. Moreover, we compute the $\alpha$-, $\beta(bp)$- and $\gamma$-duals and finally, we characterize some new 4d matrix transformation classes and complete this work with some significant results.

References

  • Altay, B., Başar F., Some new spaces of double sequences, J. Math. Anal. Appl., 309(1)(2005), 70–90.
  • Başar, F., Sever, Y., The space Lq of double sequences, Math. J. Okayama Univ., 51(2009), 149–157.
  • Cooke, R.C., Infinite Matrices and Sequence Spaces, Macmillan and Co. Limited, London, 1950.
  • Demiriz, S., Duyar,O., Domain of difference matrix of order one in some spaces of double sequences, Gulf J. Math., 3(3)(2015), 85–100.
  • Demiriz, S., Duyar,O., Domain of the generalized double Ces`aro matrix in some paranormed spaces of double sequences, Tbil. Math. J., 10(2017), 43–56.
  • Demiriz, S., İlkhan, M., Kara, E.E., Almost convergence and Euler totient matrix, Annals of Functional Analysis, 11(2020), 604–616.
  • Demiriz, S., Erdem, S., Domain of Euler-Totient matrix operator in the space Lp, Korean J. Math., 28(2)(2020), 361–378.
  • Dickson, L.E., History of the Theory of Numbers, Chelsea Publishing Company, New York, 1971.
  • Erdem, S., Demiriz, S., 4-Dimensional Euler-Totient matrix operator and some double sequence spaces, Math. Sci. Appl. E-Notes, 8(2)(2020), 110–122.
  • Erdem, S., Demiriz, S., A new RH-regular matrix derived by Jordan’s function and its domains on some double sequence spaces, Journal of Function Spaces, 2021, Article ID 5594751, 9 pages, (2021).
  • Hamilton, H.J., Transformations of multiple sequences, Duke Math. J., 2(1936), 29–60.
  • İlkhan, M., Kara, E.E., A new Banach space defined by Euler Totient matrix operator, Operators and Matrices, 13(2)(2019), 527–544.
  • İlkhan, M., Demiriz, S., Kara, E.E., A new paranormed sequence space defined by Euler totient matrix, Karaelmas Science and Engineering Journal, 9(2)(2019), 277–282.
  • Kovac, E., On φ convergence and φ density, Mathematica Slovaca, 55(2005), 329–351.
  • Lorentz, G.G., A contribution to the theory of divergent sequences, Acta Math. 80(1)(1948), 167–190.
  • M`oricz, F., Extensions of the spaces c and c0 from single to double sequences, Acta Math. Hungar., 57(1991), 129–136.
  • M`oricz, F., Rhoades, B.E., Almost convergence of double sequences and strong regularty of summability matrices, Math. Proc. Camb. Philos. Soc., 104(1988), 283–294.
  • Mursaleen, M., Almost strongly regular matrices and a core theorem for double sequences, J. Math. Anal. Appl., 293(2)(2004), 523–531.
  • Mursaleen, M., Bas.ar, F., Domain of Ces`aro mean of order one in some spaces of double sequences, Stud. Sci. Math. Hungar., 51(3)(2014), 335–356.
  • Niven, I., Zuckerman, H.S., Montgomery, H.L., An Introduction to the Theory of Numbers, (5. Edition), Wiley, New York, 1991.
  • Pringsheim, A., Zur theorie der zweifach unendlichen Zahlenfolgen, Math. Ann. 53(1900), 289–321.
  • Schaefer, H.H., Topological Vector Spaces, Graduate Texts in Mathematics, Vol. 3, 5th printing, 1986.
  • Talebi, G., Operator norms of four-dimensional Hausdorff matrices on the double Euler sequence spaces, Linear and Multilinear Algebra, 65(11)(2017), 2257–2267.
  • Tuğ, O., Four-dimensional generalized difference matrix and some double sequence spaces, J. Inequal. Appl. 2017(1)(2017), 149.
  • Tuğ, O., On almost B-summable double sequence spaces, J. Inequal. Appl. 2018(1)9, 19 pages, (2018).
  • Tuğ, O., On the characterization of some classes of four-dimensional matrices and almost B-summable double sequences, Journal of Mathematics, 2018(2018), Article ID 1826485, 7 pages.
  • Tuğ, O., Rakoˇcevi´c, V., Malkowsky, E., On the domain of the four-dimensional sequential band matrix in some double sequence spaces, Mathematics (2020), 8, 789.
  • Yeşilkayagil, M. Başar, F., Some topological properties of the spaces of almost null and almost convergent double sequences, Turkish J. Math., 40(3)(2016), 624–630.
  • Yeşilkayagil, M., Başar, F., On the characterization of a class of four dimensional matrices and Steinhaus type theorems, Kragujev. J. Math., 40(1)(2016), 35–45.
  • Yeşilkayagil, M. Başar, F., Domain of Riesz Mean in the Space Lp, Filomat, 31(4)(2017), 925–940.
  • Zeltser, M., Investigation of double sequence spaces by soft and hard analitic methods, Dissertationes Mathematicae Universtaties Tartuensis 25, Tartu University Press, Univ. of Tartu, Faculty of Mathematics and Computer Science, Tartu, 2001.
  • Zeltser, M., On conservative matrix methods for double sequence spaces, Acta Math. Hung., 95(3)(2002), 225–242.
  • Zeltser, M., Mursaleen, M., Mohiuddine, S.A., On almost conservative matrix mathods for double sequence spaces, Publ. Math. Debrecen, 75(2009), 387–399.
Year 2022, Volume: 14 Issue: 2, 271 - 280, 30.12.2022
https://doi.org/10.47000/tjmcs.1007927

Abstract

References

  • Altay, B., Başar F., Some new spaces of double sequences, J. Math. Anal. Appl., 309(1)(2005), 70–90.
  • Başar, F., Sever, Y., The space Lq of double sequences, Math. J. Okayama Univ., 51(2009), 149–157.
  • Cooke, R.C., Infinite Matrices and Sequence Spaces, Macmillan and Co. Limited, London, 1950.
  • Demiriz, S., Duyar,O., Domain of difference matrix of order one in some spaces of double sequences, Gulf J. Math., 3(3)(2015), 85–100.
  • Demiriz, S., Duyar,O., Domain of the generalized double Ces`aro matrix in some paranormed spaces of double sequences, Tbil. Math. J., 10(2017), 43–56.
  • Demiriz, S., İlkhan, M., Kara, E.E., Almost convergence and Euler totient matrix, Annals of Functional Analysis, 11(2020), 604–616.
  • Demiriz, S., Erdem, S., Domain of Euler-Totient matrix operator in the space Lp, Korean J. Math., 28(2)(2020), 361–378.
  • Dickson, L.E., History of the Theory of Numbers, Chelsea Publishing Company, New York, 1971.
  • Erdem, S., Demiriz, S., 4-Dimensional Euler-Totient matrix operator and some double sequence spaces, Math. Sci. Appl. E-Notes, 8(2)(2020), 110–122.
  • Erdem, S., Demiriz, S., A new RH-regular matrix derived by Jordan’s function and its domains on some double sequence spaces, Journal of Function Spaces, 2021, Article ID 5594751, 9 pages, (2021).
  • Hamilton, H.J., Transformations of multiple sequences, Duke Math. J., 2(1936), 29–60.
  • İlkhan, M., Kara, E.E., A new Banach space defined by Euler Totient matrix operator, Operators and Matrices, 13(2)(2019), 527–544.
  • İlkhan, M., Demiriz, S., Kara, E.E., A new paranormed sequence space defined by Euler totient matrix, Karaelmas Science and Engineering Journal, 9(2)(2019), 277–282.
  • Kovac, E., On φ convergence and φ density, Mathematica Slovaca, 55(2005), 329–351.
  • Lorentz, G.G., A contribution to the theory of divergent sequences, Acta Math. 80(1)(1948), 167–190.
  • M`oricz, F., Extensions of the spaces c and c0 from single to double sequences, Acta Math. Hungar., 57(1991), 129–136.
  • M`oricz, F., Rhoades, B.E., Almost convergence of double sequences and strong regularty of summability matrices, Math. Proc. Camb. Philos. Soc., 104(1988), 283–294.
  • Mursaleen, M., Almost strongly regular matrices and a core theorem for double sequences, J. Math. Anal. Appl., 293(2)(2004), 523–531.
  • Mursaleen, M., Bas.ar, F., Domain of Ces`aro mean of order one in some spaces of double sequences, Stud. Sci. Math. Hungar., 51(3)(2014), 335–356.
  • Niven, I., Zuckerman, H.S., Montgomery, H.L., An Introduction to the Theory of Numbers, (5. Edition), Wiley, New York, 1991.
  • Pringsheim, A., Zur theorie der zweifach unendlichen Zahlenfolgen, Math. Ann. 53(1900), 289–321.
  • Schaefer, H.H., Topological Vector Spaces, Graduate Texts in Mathematics, Vol. 3, 5th printing, 1986.
  • Talebi, G., Operator norms of four-dimensional Hausdorff matrices on the double Euler sequence spaces, Linear and Multilinear Algebra, 65(11)(2017), 2257–2267.
  • Tuğ, O., Four-dimensional generalized difference matrix and some double sequence spaces, J. Inequal. Appl. 2017(1)(2017), 149.
  • Tuğ, O., On almost B-summable double sequence spaces, J. Inequal. Appl. 2018(1)9, 19 pages, (2018).
  • Tuğ, O., On the characterization of some classes of four-dimensional matrices and almost B-summable double sequences, Journal of Mathematics, 2018(2018), Article ID 1826485, 7 pages.
  • Tuğ, O., Rakoˇcevi´c, V., Malkowsky, E., On the domain of the four-dimensional sequential band matrix in some double sequence spaces, Mathematics (2020), 8, 789.
  • Yeşilkayagil, M. Başar, F., Some topological properties of the spaces of almost null and almost convergent double sequences, Turkish J. Math., 40(3)(2016), 624–630.
  • Yeşilkayagil, M., Başar, F., On the characterization of a class of four dimensional matrices and Steinhaus type theorems, Kragujev. J. Math., 40(1)(2016), 35–45.
  • Yeşilkayagil, M. Başar, F., Domain of Riesz Mean in the Space Lp, Filomat, 31(4)(2017), 925–940.
  • Zeltser, M., Investigation of double sequence spaces by soft and hard analitic methods, Dissertationes Mathematicae Universtaties Tartuensis 25, Tartu University Press, Univ. of Tartu, Faculty of Mathematics and Computer Science, Tartu, 2001.
  • Zeltser, M., On conservative matrix methods for double sequence spaces, Acta Math. Hung., 95(3)(2002), 225–242.
  • Zeltser, M., Mursaleen, M., Mohiuddine, S.A., On almost conservative matrix mathods for double sequence spaces, Publ. Math. Debrecen, 75(2009), 387–399.
There are 33 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Sezer Erdem 0000-0001-9420-8264

Serkan Demiriz 0000-0002-4662-6020

Early Pub Date December 23, 2022
Publication Date December 30, 2022
Published in Issue Year 2022 Volume: 14 Issue: 2

Cite

APA Erdem, S., & Demiriz, S. (2022). Some Results Related to New Jordan Totient Double Sequence Spaces. Turkish Journal of Mathematics and Computer Science, 14(2), 271-280. https://doi.org/10.47000/tjmcs.1007927
AMA Erdem S, Demiriz S. Some Results Related to New Jordan Totient Double Sequence Spaces. TJMCS. December 2022;14(2):271-280. doi:10.47000/tjmcs.1007927
Chicago Erdem, Sezer, and Serkan Demiriz. “Some Results Related to New Jordan Totient Double Sequence Spaces”. Turkish Journal of Mathematics and Computer Science 14, no. 2 (December 2022): 271-80. https://doi.org/10.47000/tjmcs.1007927.
EndNote Erdem S, Demiriz S (December 1, 2022) Some Results Related to New Jordan Totient Double Sequence Spaces. Turkish Journal of Mathematics and Computer Science 14 2 271–280.
IEEE S. Erdem and S. Demiriz, “Some Results Related to New Jordan Totient Double Sequence Spaces”, TJMCS, vol. 14, no. 2, pp. 271–280, 2022, doi: 10.47000/tjmcs.1007927.
ISNAD Erdem, Sezer - Demiriz, Serkan. “Some Results Related to New Jordan Totient Double Sequence Spaces”. Turkish Journal of Mathematics and Computer Science 14/2 (December 2022), 271-280. https://doi.org/10.47000/tjmcs.1007927.
JAMA Erdem S, Demiriz S. Some Results Related to New Jordan Totient Double Sequence Spaces. TJMCS. 2022;14:271–280.
MLA Erdem, Sezer and Serkan Demiriz. “Some Results Related to New Jordan Totient Double Sequence Spaces”. Turkish Journal of Mathematics and Computer Science, vol. 14, no. 2, 2022, pp. 271-80, doi:10.47000/tjmcs.1007927.
Vancouver Erdem S, Demiriz S. Some Results Related to New Jordan Totient Double Sequence Spaces. TJMCS. 2022;14(2):271-80.