On the New Double Binomial Sequence Space

Abstract

The aim of this paper is to present the new double Binomial sequence space $\mathcal{B}_{p}^{r,s}$ which consists of all sequences whose double Binomial transforms of orders $r,s$ ($r$ and $s$ are nonzero real numbers with $r+s \neq 0$) are in the space $\mathcal{L}_p$, where $0<p<\infty$. We examine its topological and algebraic properties and inclusion relations. Furthermore, the $\alpha-$, $\beta(bp)-$ and $\gamma-$duals of the space $\mathcal{B}_{p}^{r,s}$ are determined and finally, some 4-dimensional matrix mapping classes related to this space are  characterized.

Keywords

• $\beta(bp)$- and $\gamma$-duals
• Double sequence spaces
• Binomial matrix
• Matrix domain of 4-dimensional matrices
• Matrix transformations

References

1. Altay, B., Ba\c{s}ar, F., \textit{Some new spaces of double sequences, J. Math. Anal. Appl.}, \textbf{309}(1)(2005), 70-90.
2. Ba\c{s}ar, F., Summability Theory and Its Applications, Bentham Science Publishers, e-book, Monographs, Istanbul, 2012.
3. Ba\c{s}ar, F., Sever, Y., \textit{The space $\mathcal{L}_{q}$ of double sequences}, Math. J. Okayama Univ., \textbf{51}(2009), 149-157.
4. Bi\c{s}gin, M.C., \textit{The binomial sequence spaces which include the spaces $\ell_p$ and $\ell_\infty$ and geometric properties}, Journal of Inequalities and Applications (2016):304.
5. Boss, J., Classical and Modern Methods in Summability, Oxford University Press, Newyork, 2000.
6. \c{C}apan, H., Ba\d{s}ar, F., \textit{Some paranormed difference spaces of double sequences}, Indian L. Math., \textbf{58}(3)(2016),405-427.
7. Demiriz, S., Duyar, O., \textit{Domain of the Ces\`{a}ro mean matrix in some paranormed spaces of double sequences}, Contemp. Anal. Appl. Math., \textbf{3}(2)(2015), 247-262.
8. Demiriz, S., Erdem, S., \textit{Domain of Euler-Totient Matrix Operator in the Space $\mathcal{L}_p$}, Korean J. Math., \textbf{28}(2)(2020), 361-378.

9. Hamilton, H. J., \textit{Transformations of multiple sequences}, Duke Math. J., \textbf{2}(1936), 29-60.
10. M\`{o}ricz, F., \textit{Extensions of the spaces $c$ and $c_{0}$ from single to double sequences}, Acta Math. Hungar., \textbf{57}(1991), 129-136.
11. Mursaleen, M.,\textit{ Almost strongly regular matrices and a core theorem for double sequences}, J. Math. Anal. Appl., \textbf{293}(2)(2004), 523-531.
12. Robison, G. M., \textit{Divergent double sequences and series}, Amer. Math. Soc. Trans., \textbf{28}(1926), 50-73.
13. Schaefer, H.H., Topological Vector Spaces, Graduate Texts in Matematics, Volume 3, 5th printing, 1986.
14. Talebi, G., \textit{Operator norms of four-dimensional Hausdorff matrices on the double Euler sequence spaces}, Linear and Multilinear Algebra, \textbf{65}(11)(2017), 2257-2267.
15. Tu\~{g}, O., Ba\c{s}ar, F., \textit{Four-Dimensional Generalized Difference Matrix and Some Double Sequence Spaces}, AIP Conference Proceedings, AIP Publishing LLC, \textbf{1759}(1)(2016), p.020075.
16. Tu\~{g}, O., \textit{Four-dimensional generalized difference matrix and some double sequence spaces}, J. Inequal. Appl. (1)(2017), 149.
17. Tu\~{g}, O., Rako\v{c}evi\'{c}, V., Malkowsky, E., \textit{On the Domain of the Four-Dimensional Sequential Band Matrix in Some Double Sequence Spaces}, Mathematics \textbf{789}(8)(2020), doi:10.3390/math8050789.
18. Ye\c{s}ilkayagil, M., Ba\c{s}ar, F., \textit{Domain of Riesz mean in the space $\mathcal{L}_s$}, Filomat, \textbf{31}(4)(2017), 925-940.
19. Ye\c{s}ilkayagil, M., Ba\c{s}ar, F., \textit{Domain of Riesz mean in some spaces of double sequences}, Indagationes Mathematicae, \textbf{29}(2018), 1009-1029.
20. Ye\c{s}ilkayagil, M., Ba\c{s}ar, F., \textit{Domain of Euler Mean in the Space of Absolutely $p$-Summable Double Sequences with $0
21. Zeltser, M., Investigation of double sequence spaces by soft and hard analitic methods, Dissertationes Mathematicae Universtaties Tartuensis \textbf{25}, Tartu University Press, Univ. of Tartu, Faculty of Mathematics and Computer Science, Tartu, 2001.
22. Zeltser, M., \textit{On conservative matrix methods for double sequence spaces}, Acta Math. Hung., \textbf{95}(3)(2002), 225-242.