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http://dx.doi.org/10.11568/kjm.2020.28.2.361

DOMAIN OF EULER-TOTIENT MATRIX OPERATOR IN THE SPACE 𝓛p  

Demiriz, Serkan (Department of Mathematics, Gaziosmanpasa University)
Erdem, Sezer (Farabi Anatolian Imam Hatip High School)
Publication Information
Korean Journal of Mathematics / v.28, no.2, 2020 , pp. 361-378 More about this Journal
Abstract
The most apparent aspect of the present study is to introduce a new sequence space 𝚽⋆(𝓛p) derived by double Euler-Totient matrix operator. We examine its topological and algebraic properties and give an inclusion relation. In addition to those, the α-, β(bp)- and γ-duals of the space 𝚽⋆(𝓛p) are determined and finally, some 4-dimensional matrix mapping classes related to this space are characterized.
Keywords
Euler function; $M{\ddot{o}}bius$ function; double Euler-Totient matrix; matrix domain; double sequence space; ${\alpha}$-, ${\beta}(bp)$- and ${\gamma}$-duals; matrix transformations;
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1 C.R. Adams, On non-factorable transformations of double sequences, Proc. Natl. Acad. Sci. USA 19 (5) (1933), 564-567.   DOI
2 B. Altay and F. Basar, Some new spaces of double sequences, J. Math. Anal. Appl., 309 (1) (2005), 70-90.   DOI
3 F. Basar, Summability Theory and Its Applications, Bentham Science Publishers, e-book, Monographs, Istanbul, 2012.
4 F. Basar and Y. Sever, The space ${\mathcal{L}}_q$ of double sequences, Math. J. Okayama Univ. 51 (2009), 149-157.
5 J. Boss, Classical and Modern Methods in Summability, Oxford University Press, Newyork, 2000.
6 R.C. Cooke, Infinite Matrices and Sequence Spaces, Macmillan and Co. Limited, London, 1950.
7 H. Capan and F. Basar, Some paranormed difference spaces of double sequences, Indian L. Math. 58 (3)(2016), 405-427.
8 S. Demiriz and O. Duyar, Domain of the Cesaro mean matrix in some paranormed spaces of double sequences, Contemp. Anal. Appl. Math. 3 (2) (2015), 247-262.
9 H.J. Hamilton, Transformations of multiple sequences, Duke Math. J. 2 (1936), 29-60.   DOI
10 M. Ilkhan and E. E. Kara, A New Banach Space Defined by Euler Totient Matrix Operator, Operators and Matrices 13 (2) (2019), 527-544.
11 E.E. Kara, Some topological and geometrical properties of new Banach sequence spaces, Journal of Inequalities and Applications, 2013 (38) (2013), 15 pages.
12 E.E. Kara and M. Basarir, On some Euler $B^{(m)}$ difference sequence spaces and compact operators, Journal of Mathematical Analysis and Applications, 379 (2011), 499-511.   DOI
13 E.E. Kara and S. Konca, On some new weighted Euler sequence spaces and compact operators, Mathematical Inequalities and applications 17 (2) (2014), 649-664.
14 M. Mursaleen and F. Basar, Domain of Cesaro mean of order one in some spaces of double sequences, Stud. Sci. Math. Hungar. 51 (3) (2014), 335-356.   DOI
15 E.E. Kara and M. Ilkhan, On some Banach sequence spaces derived by a new band matrix, British Journal of Mathematics and Computer Science 9 (2) (2015), 141-159.   DOI
16 E.E. Kara and M. Ilkhan, Some properties of generalized Fibonacci sequence spaces, Linear and Multilinear Algebra 64 (11) (2016), 2208-2223.   DOI
17 E. Kovac, On ${\varphi}$ convergence and ${\varphi}$ density, Mathematica Slovaca 55 (2005), 329-351.
18 F. Moricz, Extensions of the spaces c and $c_0$ from single to double sequences, Acta Math. Hungar. 57 (1991), 129-136.   DOI
19 M. Mursaleen, Almost strongly regular matrices and a core theorem for double sequences, J. Math. Anal. Appl. 293 (2) (2004), 523-531.   DOI
20 M. Mursaleen and S. A. Mohiuddine, Convergence Methods for Double Sequences and Applications, Springer, New Delhi, Heidelberg, New York, Dordrecht, London, 2014.
21 I. Niven, H.S. Zuckerman and H.L. Montgomery, An introduction to the theory of numbers, (5. Edition), Wiley, New York, 1991.
22 A. Pringsheim, Zur Theorie der zweifach unendlichen Zahlenfolgen, Math. Ann. 53, 289-321 (1900).   DOI
23 G. M. Robison, Divergent double sequences and series, Amer. Math. Soc. Trans, 28 (1926), 50-73.   DOI
24 H.H. Schaefer, Topological Vector Spaces, Graduate Texts in Matematics, Volume 3, 5th printing, 1986.
25 I. Schoenberg, The integrability of certain functions and related summability methods, The American Monthly, 66 (1959), 361-375.   DOI
26 M.Yesilkayagil and F. Basar, On the Domain of Riesz Mean in the Space ${\mathcal{L}}_s^*$, Filomat, 31 (4) (2017), 925-940.   DOI
27 M. Stieglitz and H. Tietz, Matrix transformationen von folgenraumen eine ergebnisbersicht, Mathematische Zeitschrift, 154 (1977), 1-16.   DOI
28 G. Talebi, Operator norms of four-dimensional Hausdorff matrices on the double Euler sequence spaces, Linear and Multilinear Algebra 65 (11) (2017), 2257-2267.   DOI
29 O. Tug, Four-dimensional generalized difference matrix and some double sequence spaces, J. Inequal. Appl. 2017, Articla number: 149 (2017).
30 M.Yesilkayagil and F. Basar, 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.   DOI
31 M. Zeltser, Investigation of double sequence spaces by soft and hard analitic methods, Dissertationes Mathematicae Universtaties Tartuensis 25, Tartu University Press, Univ. of Tartu, Faculty of Mathematics and Computer Science, Tartu, 2001.
32 M. Zeltser, On conservative matrix methods for double sequence spaces, Acta Math. Hung. 95 (3) (2002), 225-242.   DOI
33 M. Zeltser, M. Mursaleen and S. A. Mohiuddine, On almost conservative matrix mathods for double sequence spaces, Publ. Math. Debrecen, 75 (2009), 387-399.