Four New Sequence Spaces Obtained from the Domain of Quadruple Band Matrix Operator

Year 2023, , 294 - 311, 31.12.2023

Mustafa Cemil Bişgin

https://doi.org/10.47000/tjmcs.1274395

Abstract

In this work, we construct the sequence spaces $c_{0}(Q)$, $c(Q)$, $\ell_{\infty}(Q)$ and $\ell_{p}(Q)$ derived by the domain of quadruple band matrix, which generalizes the matrices $\Delta^{3}$, $B(r,s,t)$, $\Delta^{2}$, $B(r,s)$, $\Delta$, where $\Delta^{3}$, $B(r,s,t)$, $\Delta^{2}$, $B(r,s)$ and $\Delta$ are called third order difference, triple band, second order difference, double band and difference matrix, in turn. Also, we investigate some topological properties and some inclusion relations related to those spaces. Furthermore, we give the Schauder basis of the spaces $c_{0}(Q)$, $c(Q)$ and $\ell_{p}(Q)$, and determine $\alpha-\beta-$ and $\gamma-$duals of those spaces. Lastly, we characterize some matrix classes related to some of those spaces.

Keywords

Matrix transformations, matrix domain, Schauder basis, $\alpha-\ \ \beta- \textbf{and } \gamma-$duals, matrix classes

References

• Ahmad, Z.U., Mursaleen, M., Köthe-Toeplitz duals of some new sequence spaces and their matrix maps, Publ. Inst. Math. (Beograd), 42(1987), 57–61.
• Asma, Ç., Çolak, R., On the Köthe-Toeplitz duals of some generalized sets of difference sequences, Demonstratio Math., 33(2000), 797-803.
• Başar, F, Altay, B., On the space of sequences of p-bounded variation and related matrix mappings. Ukrainian Math. J. 55(1)(2003), 136-147.
• Başarır, M., Kayıkçı, M., On generalized Bm-Riesz difference sequence space and β-property, J. Inequal. Appl., 2009(2009), Art. ID 385029.
• Bektaş, C¸ .A., On some new generalized sequence spaces, J. Math. Anal. Appl., 277(2003), 681–688.
• Et, M., On some difference sequence spaces, Turkish J. Math., 17(1993), 18–24.
• Et, M., Çolak, R., On some generalized difference sequence spaces, Soochow J. Math., 21(4)(1995), 377–386.
• Kızmaz, H., On certain sequence spaces, Canad. Math. Bull., 24(2)(1981), 169–176.
• Kirişçi, M., Başar, F., Some new sequence spaces derived by the domain of generalized difference matrix, Comput. Math. Appl., 60(5)(2010), 1299–1309.
• Lorentz, G.G., A contribution to the theory of divergent sequences, Acta Math., 80(1948), 167–190.
• Maddox, I.J., Elements of Functional Analysis, Cambridge University Press (2nd edition), 1988.
• Malkowsky, E., Milovanovic, G.V., Rako˘cevi´c, V.,Tu˘g, O., The roots of polynomials and the operator Δ3i on the Hahn sequence space h, Comp. Appl. Math., 40(6)(2021), 222.
• Mursaleen, M., Generalized spaces of difference sequences, J. Math. Anal. Appl., 203(3)(1996), 738–745.
• Sönmez, A., Some new sequence spaces derived by the domain of the triple band matrix, Comput. Math. Appl., 62(2)(2011), 641–650.
• Stieglitz, M, Tietz, H., Matrix transformationen vonfolgenr¨aumen eine ergebnis¨ubersicht, Math. Z., 154(1977), 1–16.
• Tuğ, O., Rakocevic, V., Malkowsky, E., Domain of generalized difference operator Δ3i of order three on the Hahn sequence space h and matrix transformations, Linear and Multilinear Algebra, 70(22)(2022), 7433–7451.
• Tuğ, O., The generalized difference operator Δ3i of order three and its domain in the sequence spaces ℓ1 and bv, Journal Of Mathematics, (2022).
• Wilansky, A, Summability Throught Functional Analysis, in: North-Holland Mathematics Studies, vol. 85, Elsevier Science Publishers, Amsterdam, Newyork, Oxford, 1984.