A Study on Strongly Almost Convergent and Strongly Almost Null Binomial Double Sequence Spaces

Year 2021, Volume: 4 Issue: 4, 271 - 279, 01.12.2021

Sezer Erdem , Serkan Demiriz

https://doi.org/10.33401/fujma.987981

Cited By: 2

Abstract

The 4 dimensional (4d) binomial matrix and its domains on the classical double sequence spaces $\mathcal{L}_{p}$, $\mathcal{M}_{u}$, $\mathcal{C}_{P}$, $\mathcal{C}_{bP}$, $\mathcal{C}_{r}$, $\mathcal{C}_{f}$ and $\mathcal{C}_{f_0}$ have been described and examined by Demiriz and Erdem in the papers \cite{e.d.2}-\cite{e.d.3}. In this article, we describe two double sequence spaces with the aid of the aforementioned matrix and study some properties of these. After giving inclusion relations, we compute $\alpha-$, $\beta(bp)-$ and $\gamma-$duals and give some new matrix classes related them.

Keywords

4d binomial matrix, Double sequence space, Duals, Matrix transformations, RH regularity

References

• [1] S. Demiriz, S. Erdem, On the new double binomial sequence space, Turk. J. Math. Comput. Sci., 12(2) (2020) 101–111.
• [2] S. Demiriz, S. Erdem, Domain of binomial matrix in some spaces of double sequences, Punjab Uni. J. Math., 52(11) (2020), 65-79.
• [3] S. Erdem, S. Demiriz, Almost convergence and 4-dimensional binomial matrix, Konuralp J. Math., 8(2) (2020), 329-336.
• [4] M. Zeltser, On conservative matrix methods for double sequence spaces, Acta Math. Hung., 95(3) (2002), 225-242.
• [5] M. Zeltser, Investigation of double sequence spaces by soft and hard analytic methods, Ph.D. Thesis, Uni., Tartu, 2001.
• [6] B. Altay, F. Başar, Some new spaces of double sequences, J. Math. Anal. Appl., 309(1) (2005), 70-90.
• [7] G. Talebi, Operator norms of four-dimensional Hausdorff matrices on the double Euler sequence spaces, Linear and Multilinear Algebra, 65(11) (2017), 2257-2267.
• [8] M.Yeşilkayagil, F. Başar, Domain of Euler mean in the space of absolutely p-summable double sequences with 0 < p < 1, Anal. Theory Appl., 34(3) (2018), 241-252.
• [9] O. Tuğ, Four-dimensional generalized difference matrix and some double sequence spaces, J. Inequal. Appl., 2017(1) (2017), 1-22.
• [10] O. Tuğ, On almost B-summable double sequence spaces, J. Inequal. Appl., 2018(1) (2018), 1-19.
• [11] O. Tuğ, The spaces of B(r; s; t;u) strongly almost convergent double sequences and matrix transformations, Bull. Sci. Math., 169(2021), 102989.
• [12] M.C. Bişgin, The binomial sequence spaces which include the spaces `p and `¥ and geometric properties, J. Inequal. Appl., 2016 (2016), 304.
• [13] M.C. Bişgin, The binomial sequence spaces of nonabsolute type, J. Inequal. Appl., 2016(1) (2016), 1-16.
• [14] M.C. Bişgin, The binomial almost convergent and null sequence spaces, Commun. Fac. Sci. Uni. Ank. Series A1, 67(1) (2018), 211-224.
• [15] F. Moricz, B.E. Rhoades, Almost convergence of double sequences and strong regularity of summability matrices, Math. Proc. Camb. Philos. Soc., 104 (1988), 283-294.
• [16] M. Başarır, On the strong almost convergence of double sequences, Period. Math. Hung., 30(3) (1995), 177–181.
• [17] C.R. Adams, On non-factorable transformations of double sequences, Proc. Natl. Acad. Sci. USA, 19(5) (1933), 564-567.
• [18] P.Z. Alp, E.E. Kara, The new class Lz;p;E of s􀀀 type operators, AIMS Math., 4(3) (2019), 779-791.
• [19] F. Başar, Y. Sever, The space Lq of double sequences, Math. J. Okayama Uni., 51 (2009), 149-157.
• [20] R.C. Cooke, Infinite Matrices and Sequence Spaces, Macmillan and Co. Limited, London, 1950.
• [21] S. Erdem, S. Demiriz, A new RH-regular matrix derived by Jordan’s function and its domains on some double sequence spaces, J. Function Spaces, 2021 (2021), Article ID 5594751, 9 pages.
• [22] H. J. Hamilton, Transformations of multiple sequences, Duke Math. J., 2 (1936), 29-60.
• [23] M. İlkhan, M. A. Bayrakdar, A study on matrix domain of Riesz-Euler totient matrix in the space of p-absolutely summable sequences, Com. Adv. Math. Sci., 4(1) (2021), 14-25.
• [24] V.A. Khan, U. Tuba, On paranormed Ideal convergent sequence spaces defined by Jordan totient function, J. Inequal. Appl., 2021(1) (2021), 1-16, https://doi.org/10.1186/s13660-021-02634-7.
• [25] M. Kirisci, F. Başar, Some new sequence spaces derived by the domain of generalized difference matrix, Computers and Mathematics with Applications, 60(5) (2010), 1299–1309.
• [26] M. Mursaleen, Almost strongly regular matrices and a core theorem for double sequences, J. Math. Anal. Appl., 293(2) (2004), 523-531.
• [27] A.K. Noman, E.S. Al Yari, New general results on matrix domains of triangles in sequence spaces, Albaydha Uni. J., 3(2) (2021), 57-72.
• [28] F. Nuray, U. Ulusu, E. Dündar, Ces`aro summability of double sequences of sets, General Mathematics Notes, 25(1) (2014), 8–18.
• [29] F. Nuray, U. Ulusu, Lacunary invariant statistical convergence of double sequences of sets, Creative Math. and Info., 28(2) (2019), 143–150.
• [30] F. Nuray, E. Dündar, U. Ulusu, Wijsman statistical convergence of double sequences of set, Iranian J. Math. Sci. Info., 16(1) (2021), 55–64.
• [31] A. Pringsheim, Zur Theorie der zweifach unendlichen Zahlenfolgen, Math. Ann., 53 (1900), 289-321.
• [32] G. M. Robison, Divergent double sequences and series, Amer. Math. Soc. Trans., 28 (1926), 50-73.
• [33] O. Tuğ, V. Rakocevic, E. Malkowsky, On the domain of the four-dimensional sequential band matrix in some double sequence Spaces, Mathematics, 8(5)(2020), 789.
• [34] O. Tuğ, The spaces of B(r; s; t;u) strongly almost convergent double sequences and matrix transformations, Bull. Sci. Math., 169 (2021), 102989.
• [35] T. Yaying, B. Hazarika, On sequence spaces generated by binomial difference operator of fractional order, Math. Slovaca, 69(4) (2019), 901–918.
• [36] M.Yeşilkayagil, F. Başar, Some topological properties of the spaces of almost null and almost convergent double sequences, Turk. J. Math., 40(3) (2016), 624-630.
• [37] M.Yeşilkayagil, F. Başar, On the characterization of a class of four-dimensional matrices and Steinhaus type theorems, Kragujevac J. Math., 40(1)(2016), 35-45.
• [38] M.Yeşilkayagil, F. Başar, Domain of Riesz mean in the space Lp, Filomat, 31(4) (2017), 925-940.
• [39] M. Zeltser, M. Mursaleen, S. A. Mohiuddine, On almost conservative matrix methods for double sequence spaces, Publ. Math. Debrecen, 75 (2009), 387-399.