Research Article

In this study, a special lower triangular matrix derived by combining Riesz matrix and Euler totient matrix is used to construct new Banach spaces. $\alpha-$,$\beta-$,$\gamma-$duals of the resulting spaces are obtained and some matrix operators are characterized. Finally by the aid of measure of non-compactness, the conditions for which a matrix operator on these spaces is compact are determined.

sequence spaces $\alpha-$,$\beta-$,$\gamma-$duals Matrix mappings Hausdorff measure of non-compactness Compact operators

- [1] B. Altay, F. Basar, M. Mursaleen, On the Euler sequence spaces which include the spaces p and I, Inform. Sci., 176(10) (2006), 1450-1462.
- [2] F. Bas¸ar, B. Altay, On the space of sequences of p-bounded variation and related matrix mappings, Ukrainian Math. J., 55(1) (2003), 136-147.
- [3] F. Basar, Summability Theory and Its Applications, Bentham Science Publishers, Istanbul, 2012.
- [4] M. Ilkhan, Certain geometric properties and matrix transformations on a newly introduced Banach space, Fundam. J. Math. Appl., 3(1) (2020), 45-51.
- [5] E. E. Kara, Some topological and geometrical properties of new Banach sequence spaces, J. Inequal. Appl., 2013(38) (2013), 15 pages.
- [6] M. Kirisci, F. Basar, Some new sequence spaces derived by the domain of generalized difference matrix, Comput. Math. Appl., 60 (2010), 1299-1309.
- [7] M. Kirisci, Riesz type integrated and differentiated sequence spaces, Bull. Math. Anal. Appl., 7(2) (2015), 14-27.
- [8] S.A. Mohiuddine, A. Alotaibi, Weighted almost convergence and related infinite matrices, J. Inequal. Appl., 2018(15) (2018), 10 pages.
- [9] M. Mursaleen, A.K. Noman, On some new difference sequence spaces of non-absolute type, Math. Comput. Modelling, 52(3-4) (2010), 603-617.
- [10] T. Yaying, B. Hazarika, On sequence spaces defined by the domain of a regular Tribonacci matrix, Math. Slovaca, 70(3) (2020), 697-706.
- [11] P. Zengin Alp, A new paranormed sequence space defined by Catalan conservative matrix, Math. Methods Appl. Sci., doi: https://doi.org/10.1002/mma.6530.
- [12] T. Yaying, B. Hazarika, On sequence spaces generated by binomial difference operator of fractional order, Math. Slovaca, 69(4) (2019), 901-918.
- [13] T. Yaying, B. Hazarika, M. Mursaleen, On sequence space derived by the domain of q-Cesaro matrix in `p space and the associated operator ideal, J. Math. Anal. Appl., 493(1) (2021), 124453.
- [14] T. Yaying, B. Hazarika, S.A. Mohiuddine, M. Mursaleen, K.J. Ansari, Sequence spaces derived by the triple band generalized Fibonacci difference operator, Adv. Diff. Equ., 2020 (2020), 639.
- [15] T. Yaying, Paranormed Riesz difference sequence spaces of fractional order, Kragujevac J. Math., 46(2) (2022), 175-191.
- [16] E. Malkowsky, V. Rakocevic, An introduction into the theory of sequence spaces and measure of noncompactness, Zb. Rad. (Beogr.), 17 (2000), 143-234.
- [17] V. Rakocevic, Measures of noncompactness and some applications, Filomat, 12 (1998), 87-120.
- [18] M. Basarır, E. E. Kara, On some difference sequence spaces of weighted means and compact operators, Ann. Funct. Anal. 2 (2011), 114-129.
- [19] M. Basarır, E. E. Kara, On the B-difference sequence space derived by generalized weighted mean and compact operators, J. Math. Anal. Appl. 391 (2012), 67-81.
- [20] M. Bas¸arır, E. E. Kara, On compact operators on the Riesz Bm-difference sequence spaces II, Iran. J. Sci. Technol. Trans. A Sci., 36 (2012), 371-376.
- [21] E. E. Kara, M. Basarır, On compact operators and some Euler B(m)-difference sequence spaces, J. Math. Anal. Appl., 379(2) (2011), 499-511.
- [22] M. Mursaleen, A. K. Noman, Compactness by the Hausdorff measure of noncompactness, Nonlinear Anal., 73(8) (2010), 2541-2557.
- [23] M. Mursaleen, A. K. Noman, The Hausdorff measure of noncompactness of matrix operators on some BK spaces, Oper. Matrices, 5 (2011), 473-486.
- [24] T. Yaying, A. Das, B. Hazarika, P. Baliarsingh, Compactness of binomial difference operator of fractional order and sequence spaces, Rend. Circ. Mat. Palermo (II) Ser., 68 (2019), 459-476.
- [25] M. Ilkhan, E. E. Kara, A new Banach space defined by Euler totient matrix operator, Oper. Matrices, 13(2) (2019), 527-544.
- [26] S. Demiriz, S. Erdem, Domain of Euler-totient matrix operator in the space Lp, Korean J. Math., 28(2) (2020), 361-378.
- [27] S. Demiriz, M. Ilkhan, E. E. Kara, Almost convergence and Euler totient matrix, Ann. Funct. Anal., 11(3) (2020), 604-616.
- [28] S. Erdem, S. Demiriz, 4-dimensional Euler-totient matrix operator and some double sequence spaces, Math. Sci. Appl. E-Notes, 8(2) (2020), 110-122.
- [29] G. C. Hazar Gulec, Merve Ilkhan, A new paranormed series space using Euler totient means and some matrix transformations, Korean J. Math., 28(2) (2020), 205-221.
- [30] G. C. Hazar Gulec, Merve Ilkhan, A new characterization of absolute summability factors, Commun. Optim. Theory, 2020 (2020), Article ID 15, 1-11.
- [31] M. Ilkhan, Matrix domain of a regular matrix derived by Euler totient function in the spaces c0 and c, Mediterr. J. Math., 17 (2020), Article number 27.
- [32] M. Ilkhan, G. C. Hazar Gulec, A study on absolute Euler totient series space and certain matrix transformations, Mugla J. Sci. Technol., 6(1) (2020), 112-119.
- [33] M. Ilkhan, S. Demiriz, E. E. Kara, A new paranormed sequence space defined by Euler totient matrix, Karaelmas Sci. Eng. J., 9(2) (2019), 277-282.
- [34] B. Altay, F. Basar, Some paranormed Riesz sequence spaces of non-absolute type, Southeast Asian Bull. Math., 30(4) (2006), 591-608.
- [35] M. Stieglitz, H. Tietz, Matrix transformationen von folgenraumen eine ergebnisubersicht, Math. Z., 154 (1977), 1-16.

Year 2021,
Volume: 4 Issue: 1, 14 - 25, 29.03.2021
### Abstract

### References

- [1] B. Altay, F. Basar, M. Mursaleen, On the Euler sequence spaces which include the spaces p and I, Inform. Sci., 176(10) (2006), 1450-1462.
- [2] F. Bas¸ar, B. Altay, On the space of sequences of p-bounded variation and related matrix mappings, Ukrainian Math. J., 55(1) (2003), 136-147.
- [3] F. Basar, Summability Theory and Its Applications, Bentham Science Publishers, Istanbul, 2012.
- [4] M. Ilkhan, Certain geometric properties and matrix transformations on a newly introduced Banach space, Fundam. J. Math. Appl., 3(1) (2020), 45-51.
- [5] E. E. Kara, Some topological and geometrical properties of new Banach sequence spaces, J. Inequal. Appl., 2013(38) (2013), 15 pages.
- [6] M. Kirisci, F. Basar, Some new sequence spaces derived by the domain of generalized difference matrix, Comput. Math. Appl., 60 (2010), 1299-1309.
- [7] M. Kirisci, Riesz type integrated and differentiated sequence spaces, Bull. Math. Anal. Appl., 7(2) (2015), 14-27.
- [8] S.A. Mohiuddine, A. Alotaibi, Weighted almost convergence and related infinite matrices, J. Inequal. Appl., 2018(15) (2018), 10 pages.
- [9] M. Mursaleen, A.K. Noman, On some new difference sequence spaces of non-absolute type, Math. Comput. Modelling, 52(3-4) (2010), 603-617.
- [10] T. Yaying, B. Hazarika, On sequence spaces defined by the domain of a regular Tribonacci matrix, Math. Slovaca, 70(3) (2020), 697-706.
- [11] P. Zengin Alp, A new paranormed sequence space defined by Catalan conservative matrix, Math. Methods Appl. Sci., doi: https://doi.org/10.1002/mma.6530.
- [12] T. Yaying, B. Hazarika, On sequence spaces generated by binomial difference operator of fractional order, Math. Slovaca, 69(4) (2019), 901-918.
- [13] T. Yaying, B. Hazarika, M. Mursaleen, On sequence space derived by the domain of q-Cesaro matrix in `p space and the associated operator ideal, J. Math. Anal. Appl., 493(1) (2021), 124453.
- [14] T. Yaying, B. Hazarika, S.A. Mohiuddine, M. Mursaleen, K.J. Ansari, Sequence spaces derived by the triple band generalized Fibonacci difference operator, Adv. Diff. Equ., 2020 (2020), 639.
- [15] T. Yaying, Paranormed Riesz difference sequence spaces of fractional order, Kragujevac J. Math., 46(2) (2022), 175-191.
- [16] E. Malkowsky, V. Rakocevic, An introduction into the theory of sequence spaces and measure of noncompactness, Zb. Rad. (Beogr.), 17 (2000), 143-234.
- [17] V. Rakocevic, Measures of noncompactness and some applications, Filomat, 12 (1998), 87-120.
- [18] M. Basarır, E. E. Kara, On some difference sequence spaces of weighted means and compact operators, Ann. Funct. Anal. 2 (2011), 114-129.
- [19] M. Basarır, E. E. Kara, On the B-difference sequence space derived by generalized weighted mean and compact operators, J. Math. Anal. Appl. 391 (2012), 67-81.
- [20] M. Bas¸arır, E. E. Kara, On compact operators on the Riesz Bm-difference sequence spaces II, Iran. J. Sci. Technol. Trans. A Sci., 36 (2012), 371-376.
- [21] E. E. Kara, M. Basarır, On compact operators and some Euler B(m)-difference sequence spaces, J. Math. Anal. Appl., 379(2) (2011), 499-511.
- [22] M. Mursaleen, A. K. Noman, Compactness by the Hausdorff measure of noncompactness, Nonlinear Anal., 73(8) (2010), 2541-2557.
- [23] M. Mursaleen, A. K. Noman, The Hausdorff measure of noncompactness of matrix operators on some BK spaces, Oper. Matrices, 5 (2011), 473-486.
- [24] T. Yaying, A. Das, B. Hazarika, P. Baliarsingh, Compactness of binomial difference operator of fractional order and sequence spaces, Rend. Circ. Mat. Palermo (II) Ser., 68 (2019), 459-476.
- [25] M. Ilkhan, E. E. Kara, A new Banach space defined by Euler totient matrix operator, Oper. Matrices, 13(2) (2019), 527-544.
- [26] S. Demiriz, S. Erdem, Domain of Euler-totient matrix operator in the space Lp, Korean J. Math., 28(2) (2020), 361-378.
- [27] S. Demiriz, M. Ilkhan, E. E. Kara, Almost convergence and Euler totient matrix, Ann. Funct. Anal., 11(3) (2020), 604-616.
- [28] S. Erdem, S. Demiriz, 4-dimensional Euler-totient matrix operator and some double sequence spaces, Math. Sci. Appl. E-Notes, 8(2) (2020), 110-122.
- [29] G. C. Hazar Gulec, Merve Ilkhan, A new paranormed series space using Euler totient means and some matrix transformations, Korean J. Math., 28(2) (2020), 205-221.
- [30] G. C. Hazar Gulec, Merve Ilkhan, A new characterization of absolute summability factors, Commun. Optim. Theory, 2020 (2020), Article ID 15, 1-11.
- [31] M. Ilkhan, Matrix domain of a regular matrix derived by Euler totient function in the spaces c0 and c, Mediterr. J. Math., 17 (2020), Article number 27.
- [32] M. Ilkhan, G. C. Hazar Gulec, A study on absolute Euler totient series space and certain matrix transformations, Mugla J. Sci. Technol., 6(1) (2020), 112-119.
- [33] M. Ilkhan, S. Demiriz, E. E. Kara, A new paranormed sequence space defined by Euler totient matrix, Karaelmas Sci. Eng. J., 9(2) (2019), 277-282.
- [34] B. Altay, F. Basar, Some paranormed Riesz sequence spaces of non-absolute type, Southeast Asian Bull. Math., 30(4) (2006), 591-608.
- [35] M. Stieglitz, H. Tietz, Matrix transformationen von folgenraumen eine ergebnisubersicht, Math. Z., 154 (1977), 1-16.

There are 35 citations in total.

Primary Language | English |
---|---|

Subjects | Mathematical Sciences |

Journal Section | Articles |

Authors | |

Publication Date | March 29, 2021 |

Submission Date | December 22, 2020 |

Acceptance Date | March 12, 2021 |

Published in Issue | Year 2021 Volume: 4 Issue: 1 |