4-Dimensional Euler-Totient Matrix Operator and Some Double Sequence Spaces

Year 2020, Volume: 8 Issue: 2, 110 - 122, 15.10.2020

Sezer Erdem , Serkan Demiriz

https://doi.org/10.36753/mathenot.733364

Cited By: 3

Abstract

Our main purpose in this study is to define
the 4-dimensional Euler-Totient matrix operator and to investigate the matrix domains
of this matrix on the classical double sequence spaces $\mathcal{M}_{u}$, $\mathcal{C}_{p}$, $\mathcal{C}_{bp}$ and $\mathcal{C}_{r}$.
Besides these, we examine their topological and algebraic properties and give inclusion relations about the new spaces.
Also, the $\alpha-$, $\beta(\vartheta)-$ and $\gamma-$duals of these spaces are determined and finally, some matrix classes are characterized.

Keywords

Euler function, Möbius function, 4-dimensional Euler-Totient matrix operator, Double sequence spaces

References

• \bibitem{adams} C.R. Adams, On non-factorable transformations of double sequences, Proc. Natl. Acad. Sci. USA, 19(5) (1933), 564-567.
• \bibitem{bafb1} B. Altay and F. Ba\c{s}ar, Some new spaces of double sequences, J. Math. Anal. Appl., \textbf{309}(1) (2005), 70-90.
• \bibitem{fb} F. Ba\d{s}ar, Summability Theory and Its Applications, Bentham Science Publishers, e-book, Monographs, Istanbul, 2012.
• \bibitem{fbys} F. Ba\c{s}ar and Y. Sever, The space $\mathcal{L}_{q}$ of double sequences, Math. J. Okayama Univ., \textbf{51} (2009), 149-157.
• \bibitem{boos}J. Boss, Classical and Modern Methods in Summability, Oxford University Press, Newyork, 2000.
• \bibitem{cooke}R.C. Cooke, Infinite Matrices and Sequence Spaces, Macmillan and Co. Limited, London, 1950.
• \bibitem{cakan}C. \c{C}akan, B. Altay and M. Mursaleen, The $\sigma$-convergence and $\sigma$-core of double sequences, Appl. Math. Lett. \textbf{19} (2006), 387-399.
• \bibitem{demiriz1}S. Demiriz and O. Duyar, Domain of the Ces\`{a}ro mean matrix in some paranormed spaces of double sequences, Contemp. Anal. Appl. Math., 3(2) (2015), 247-262.
• \bibitem{hjh} H.J. Hamilton, Transformations of multiple sequences, Duke Math. J., \textbf{2} (1936), 29-60.
• \bibitem{eek1} M. İlkhan and E. E. Kara, A New Banach Space Defined by Euler Totient Matrix Operator, Operators and Matrices, \textbf{13}(2) (2019), 527-544.
• \bibitem{eek5} E.E. Kara and M. İlkhan, On some Banach sequence spaces derived by a new band matrix, British Journal of Mathematics and Computer Science, \textbf{9}(2) (2015), 141-159.
• \bibitem{eek6} E.E. Kara and M. İlkhan, Some properties of generalized Fibonacci sequence spaces, Linear and Multilinear Algebra, \textbf{64}(11) (2016), 2208-2223.
• \bibitem{kovac} E. Kovac, On $\varphi$ convergence and $\varphi$ density, Mathematica Slovaca, \textbf{55}(2005), 329-351.
• \bibitem{fm} F. M\`{o}ricz, Extensions of the spaces $c$ and $c_{0}$ from single to double sequences, Acta Math. Hungar., \textbf{57} (1991), 129-136.
• \bibitem{mm1} M. Mursaleen, Almost strongly regular matrices and a core theorem for double sequences, J. Math. Anal. Appl., \textbf{293}(2) (2004), 523-531.
• \bibitem{mmfb} M. Mursaleen and F. Ba\d{s}ar, Domain of Ces\`{a}ro mean of order one in some spaces of double sequences, Stud. Sci. Math. Hungar., \textbf{51}(3) (2014), 335-356.
• \bibitem{mmm} M. Mursaleen and S. A. Mohiuddine, Convergence Methods for Double Sequences and Applications, Springer, New Delhi, Heidelberg, New York, Dordrecht, London, 2014.
• \bibitem{niven} I. Niven, H.S. Zuckerman and H.L. Montgomery, An introduction to the theory of numbers, (5. Edition),Wiley, New York, 1991.
• \bibitem{pring}A. Pringsheim, Zur Theorie der zweifach unendlichen Zahlenfolgen, Math. Ann. 53, 289-321(1900).
• \bibitem{gmr} G. M. Robison, Divergent double sequences and series, Amer. Math. Soc. Trans., \textbf{28} (1926), 50-73.
• \bibitem{hhs} H.H. Schaefer, Topological Vector Spaces, Graduate Texts in Matematics, Volume 3, 5th printing, 1986.
• \bibitem{is} I. Schoenberg, The integrability of certain functions and related summability methods, The American Monthly, \textbf{66} (1959), 361-375.
• \bibitem{ms} M. Stieglitz and H. Tietz, Matrix transformationen von folgenraumen eine ergebnisbersicht, Mathematische Zeitschrift, \textbf{154} (1977), 1-16.
• \bibitem{talebi} G. Talebi, Operator norms of four-dimensional Hausdorff matrices on the double Euler sequence spaces, Linear and Multilinear Algebra, 65(11) (2017), 2257-2267.
• \bibitem{tug1}O. Tu\~{g}, Four-dimensional generalized difference matrix and some double sequence spaces. J. Inequal. Appl. 2017(1), 149 (2017).
• \bibitem{YB6}M.Yeşilkayagil and F. Başar, Four dimensional dual and dual of some new sort summability methods, Contemp.Anal.Appl.Math.\textbf{3}(1),(2015),pp.13-29.
• \bibitem{YB4}M.Yeşilkayagil and F. Başar, Mercerian theorem for four dimensional matrices, Commun. Fac. Sci. Univ. Ank. Ser., A1\textbf{65}(1),(2016), pp. 147-155.
• \bibitem{YB5}M.Yeşilkayagil and F. Başar, On the characterization of a class of four dimensional matrices and Steinhaus type theorems, Kragujev. J. Math. \textbf{40}(1)(2016), pp. 35-45.
• \bibitem{YB1} M.Yeşilkayagil and F. Başar, Domain of Riesz Mean in the Space $\mathcal{L}_{p}$, Filomat, \emph{31}(4) (2017), 925-940.
• \bibitem{zel1} M. Zeltser, 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.
• \bibitem{zel2} M. Zeltser, On conservative matrix methods for double sequence spaces, Acta Math. Hung., \textbf{95}(3) (2002), 225-242.
• \bibitem{mzmmsam} M. Zeltser, M. Mursaleen and S. A. Mohiuddine, On almost conservative matrix mathods for double sequence spaces, Publ. Math. Debrecen, \textbf{75} (2009), 387-399.