A New Aspect for Some Sequence Spaces Derived Using the Domain of the Matrix $\widehat{\widehat{B}}$

Abstract

This study serves for analysing algebraic and topological characteristics of the sequence spaces $X(\widehat{\widehat{B}}(r,s))$ constituted by using non-zero real number $r$ and $s$, where $X$ denotes arbitrary of the classical sequence spaces $\ell_{\infty}, c, c_{0} $ and $\ell_{p}$ $(1<p<\infty)$ of bounded, convergent, null and absolutely $p$-summable sequences, respectively and $X(\widehat{\widehat{B}})$ also is the domain of the matrix $\widehat{\widehat{B}}(r,s)$ in the sequence space $X$. Briefly, the $\beta$- and $\gamma$-duals of the space $X(\widehat{\widehat{B}})$ are computed, and Schauder bases for the spaces $c(\widehat{\widehat{B}})$, $c_{0}(\widehat{\widehat{B}})$ and $\ell_{p}(\widehat{\widehat{B}})$ are determined, and some algebraic and topological properties of the spaces $c_{0}(\widehat{\widehat{B}})$, $\ell_{1}(\widehat{\widehat{B}})$ and $\ell_{p}(\widehat{\widehat{B}})$ are studied. Additionally, it is observed that all these spaces have some remarkable features, including the classes $(X_{1}(\widehat{\widehat{B}})$: $X_{2})$ and $(X_{1}(\widehat{\widehat{B}})
: X_{2}(\widehat{\widehat{B}}))$ of infinite matrices which are characterized, in which $X_{1}\in\{ \ell_{\infty},c,c_{0},\ell_{p},\ell_{1}\}$ and $X_{2}\in\{\ell_{\infty},c,c_{0},\ell_{1}\}$.

Keywords

• Matrix domain of a sequence space
• Schauder basis
• $\beta-$ and $% \gamma-$duals and matrix transformations

References

1. [1] F. Başar, Summability Theory and Its Applications, Bentham Science Publishers, İstanbul, 2012, ISBN: 978-1-60805-252-3.
2. [2] F. Başar & H. Dutta, Summable Spaces and Their Duals, Matrix Transformations and Geometric Properties, CRC Press, Taylor & Francis Group, Monographs and Research Notes in Mathematics, Boca Raton · London · New York, 2020. ISBN: 978-0-8153-5177-1.
3. [3] M. Mursaleen, F. Başar, Sequence Spaces: Topics in Modern Summability Theory, CRC Press, Taylor & Francis Group, Series: Mathematics and Its Applications, Boca Raton · London · New York, 2020.
4. [4] M. Mursaleen, Applied Summability Methods, Springer Briefs, 2014.
5. [5] B. de Malafosse, E.Malkowsky, and V. Rakocevic, Operators Between Sequence Spaces and Applications, Springer Nature Singapore, 152 Beach Road, Singapore 18972, Singapore.
6. [6] H. Kızmaz, On certain sequence spaces, Canad. Math. Bull. 24(2) (1981), 169-176.
7. [7] M. Et, On some difference sequence spaces, Turk. J. Math. 17 (1993), 18-24.
8. [8] M. A. Sarıgöl, On difference sequence spaces, J. Karadeniz Tech. Uni. Fac. Arts Sci. Ser. Math.-Phys., 10 (1987), 63-71.

9. [9] Z. U. Ahmad, M. Mursaleen, Köthe-Toeplitz duals of some new sequence spaces and their matrix maps, Publ. Inst. Math. (Beograd) 42 (1987), 57-61.
10. [10] E. Malkowsky, Absolute and ordinary Köthe-Toeplitz duals of some sets of sequences and matrix transformations, Publ. Inst. Math., (Beograd)(NS), 46(60) (1989), 97-103.
11. [11] B. Choudhary, S. K. Mishra, A note on certain sequence spaces, J. Anal., 1 (1993), 139-148.
12. [12] S. K. Mishra, Matrix maps involving certain sequence spaces, Indian J. Pure Appl. Math., 24(2) (1993), 125-132.
13. [13] M. Mursaleen, A. K. Gaur, A. H. Saifi,Some new sequence spaces and their duals and matrix transformations, Bull. Calcutta Math. Soc., 88(3) (1996), 207-212.
14. [14] C. Gnanaseelan, P. D. Srivastava, The a􀀀, b􀀀 and g􀀀duals of some generalised difference sequence spaces, Indian J. Math., 38(2) (1996), 111-120.
15. [15] E. Malkowsky, A note on the Köthe-Toeplitz duals of generalized sets of bounded and convergent difference sequences, J. Anal., 4 (1996), 981-91.
16. [16] A. K. Gaur, M. Mursaleen, Difference sequence spaces, Int. J. Math. Math. Sci., 21(4) (1998), 701-706.
17. [17] E. Malkowsky, M. Mursaleen, Qamaruddin, Generalized sets of difference sequences, their duals and matrix transformations, In: Sequence Spaces and App. Narosa, New Delhi, 1999, pp. 68-83.
18. [18] Ç . Asma, R. Ç olak, On the Köthe-Toeplitz duals of some generalized sets of difference sequences, Demonstratio Math., 33 (2000), 797-803.
19. [19] E. Malkowsky, M. Mursaleen, Some matrix transformations between the difference sequence spaces Dc0(p), Dc(p) and D`¥(p), Filomat, 15 (2001), 353-363.
20. [20] M. Kirişçi, F. Başar, Some new sequence spaces derived by the domain of generalized difference matrix, Comput. Math. Appl., 60(5) (2010), 1299-1309.
21. [21] A. Sönmez, A. Some new sequence spaces derived by the domain of the triple band matrix, Comput. Math. Appl., 62(2) (2011), 641-650.
22. [22] M. Candan, Domain of the double sequential band matrix in the classical sequence spaces, J. Inequal. Appl., 281(1) (2012), 1-15.
23. [23] M. Candan, Almost convergence and double sequential band matrix, Acta Math. Sci. Ser. B, 34(2) (2014) 354-366.
24. [24] M. Candan, Domain of the double sequential band matrix in the spaces of convergent and null sequences, Adv. Differ. Equ., 163(1) (2014), 1-18.
25. [25] M. Stieglitz, H. Tietz, Matrix transformationen von folgenr¨aumen eine ergebnis¨ubersicht, Math. Z., 154 (1977), 1-16.
26. [26] K.-G. Grosse-Erdmann, On `1-invariant sequence spaces, J. Math. Anal. Appl., 262 (2001), 112-132.
27. [27] B. Altay, F. Başar, Certain topological properties and duals of the matrix domain of a triangle matrix in a sequence space, J. Math. Anal. Appl., 336(1) (2007), 632-645.
28. [28] F. Başar, B. Altay, On the space of sequences of p-bounded variation and related matrix mappings, Ukrainian Math. J., 55(1) (2003), 136-147.
29. [29] F. Yaşar, K. Kayaduman, A different study on the spaces of generalized Fibonacci difference bs and cs spaces sequence, Symmetry, 10 (2018), 274, doi:10.3390/sym10070274.
30. [30] K. Kayaduman, F. Yaşar, On domain of Nörlund matrix Mathematics, 6 (2018), 268, doi:10.3390/math6110268.
31. [31] F. Yaşar, K. Kayaduman, On the domain of the Fibonacci difference matrix, Mathematics, 7 (2019), 204, doi:10.3390/math7020204.
32. [32] K. Kayaduman, F. Yaşar, A. Çetin, On some inequalities and sB(r;s)-conservative matrices, J. Inequal. Speci. Func., 9(2) (2018), 82-91.
33. [33] M. Candan, K. Kayaduman, Almost convergent sequence space derived by generalized Fibonacci matrix and Fibonacci core, British J. Math. \& Computer Sci., 7 (2015), 150-167.
34. [34] M. İlkhan, E. E. Kara, A new Banach space defined by Euler totient matrix operator, Oper. Matrices, 13(2) (2019), 527-544.
35. [35] E. E. Kara, Some topological and geometrical properties of new Banach sequence spaces, J. Inequal. Appl., 38 (2013), https://doi.org/10.1186/1029-242X-2013-38.
36. [36] E. E. Kara, M. İlkhan, Some properties of generalized Fibonacci sequence spaces, Linear Multilinear Algebra, 64(11) (2016), 2208–2223, 1145626.
37. [37] M. Kirişçi, The application domain of infinite matrices with algorithms, Uni. J. Math. Appl., 1(1) (2018), 1-9.
38. [38] F. Başar, M. Kirişçi, Almost convergence and generalized difference matrix, Comput. Math. Appl., 61(3) (2011), 602-611.