A STUDY ON ABSOLUTE EULER TOTIENT SERIES SPACE AND CERTAIN MATRIX TRANSFORMATIONS

Yıl 2020, Cilt: 6 Sayı: 1, 112 - 119, 30.06.2020

Merve İlkhan , G. Canan Hazar Güleç

https://doi.org/10.22531/muglajsci.727517

Cited By: 1

Öz

Recently, many authors have focused on the studies related to sequence and series spaces. In the literature the simple and fundamental method is to construct new sequence and series spaces by means of the matrix domain of triangular matrices on the classical sequence spaces. Based on this approach, in this study, we introduce a new series space |ϕ_z |_p as the set of all series summable by absolute summability method |Φ,z_n |_p, where Φ=(ϕ_nk ) denotes Euler totient matrix, z=(z_n ) is a sequence of non-negative terms and p≥1. Also, we show that the series space |ϕ_z |_p is linearly isomorphic to the space of all p- absolutely summable sequences l_p for p≥1. Moreover, we determine some topological properties and α, β and γ-duals of this space and give Schauder basis for the space |ϕ_z |_p. Finally, we characterize the classes of the matrix operators from the space |ϕ_z |_p to the classical spaces l_∞,c,c_0,l_1 for 1≤p<∞ and vice versa.

Anahtar Kelimeler

Absolute series spaces, Matrix operators, BK spaces

Kaynakça

• Referans1 Altay, B. Başar, F. and Malkowsky, E., "Matrix transformations on some sequence spaces related to strong Cesàro summability and boundedness", Appl. Math. Comput, Vol. 211 No.2, 255-264, 2009.
• Referans2 Başarır, M. and Kara, E.E., "On some difference sequence spaces of weighted means and compact operators", Ann. Funct. Anal. Vol.2, 114-129, 2011.
• Referans3 Başarır, M. and Kara E.E., "On compact operators on the Riesz B^m-difference sequence spaces", Iran. J. Sci. Technol. Trans. A Sci. Vol.35, 279-285, 2011.
• Referans4 Bektaş, Ç.A., Et, M. and Çolak, R., "Generalized difference sequence spaces and their dual spaces", J. Math. Anal. Appl., Vol.292 No.2, 423-432, 2004.
• Referans5 Bor, H., "On |N,p_n |_k summability factors of infinite series", Tamkang J. Math., Vol.16, 13–20, 1985.
• Referans6 Borwein, D. and Cass, F.P., "Strong Nörlund summability", Math. Zeitschr., Vol.103, 94–111, 1968.
• Referans7 Et, M., "On some difference sequence spaces" Turk. J. Math., Vol.17, 18-24, 1993. Referans8 Flett, T.M., "On an extension of absolute summability and some theorems of Littlewood and Paley", Proc. London Math. Soc., Vol.7, 113-141, 1957.
• Referans9 Hazar, G.C. and Sarıgöl, M.A., "On absolute Nörlund spaces and matrix operators", Acta Math. Sin. (Engl. Ser.), Vol.34 No.5, 812-826, 2018.
• Referans10 Hazar, G.C. and Sarıgöl, M.A., "Absolute Cesàro series spaces and matrix operators", Acta App. Math., Vol.154, 153–165, 2018.
• Referans11 İlkhan, M. and Kara E.E., "A new Banach space defined by Euler totient matrix operator", Oper. Matrices, Vol.13 No.2, 527-544, 2019.
• Referans12 Kara, E.E. and Başarır, M., "On compact operators and some Euler B(m) difference sequence spaces", J. Math. Anal. Appl. Vol.379 No.2, 499-511, 2011.
• Referans13 Kara, E.E. and İlkhan, M., "Some properties of generalized Fibonacci sequence spaces", Linear Multilinear Algebra. Vol.64 No.11, 2208-2223, 2016.
• Referans14 Kirişçi, M. and Başar, F., "Some new sequence spaces derived by the domain of generalized difference matrix", Comput. Math. Appl., No.60, 1299-1309, 2010.
• Referans15 Kirişçi, M., "Riesz type integrated and differentiated sequence spaces", Bull. Math. Anal. Appl. Vol.7 No.2, 14-27, 2015.
• Referans16 Maddox, I.J., Elements of functinal analysis, Cambridge University Press, London,New York, 1970.
• Referans17 Mursaleen, M. and Noman, AK., "Compactness by the Hausdorff measure of noncompactness", Nonlinear Anal. Vol.73 No.8, 2541-2557, 2010.
• Referans18 Mursaleen, M. and Noman AK., "Applications of the Hausdorff measure of noncompactness in some sequence spaces of weighted means", Comput. Math. Appl. Vol.60 No.5, 1245-1258, 2010.
• Referans19 Mursaleen, M. and Noman A.K., "On the spaces of λ-convergent and bounded sequences", Thai J. Math. Vol.8 No.2, 311-329, 2012.
• Referans20 Mohiuddine, S.A., "An application of almost convergence in approximation theorems", Appl. Math. Lett. Vol.24, 1856-1860, 2011.
• Referans21 Mohiuddine, S.A. and Alotaibi, A., "Weighted almost convergence and related infinite matrices", J. Inequal. Appl., Article Number.15, 2018.
• Referans22 Sarıgöl, M.A., "Spaces of Series Summable by Absolute Cesàro and Matrix Operators", Comm. Math Appl. Vol.7 No.1, 11-22, 2016.
• Referans23 Sarıgöl, M.A., "Extension of Mazhar’s theorem on summability factors", Kuwait J. Sci. Vol.42 No.3, 28-35, 2015.
• Referans24 Sarıgöl, M.A., "On the local properties of factored Fourier series", Appl. Math. Comp., Vol.216, 3386-3390, 2010.
• Referans25 Schoenberg, I., "The integrability of certain functions and related summability methods", Amer. Math. Monthly, Vol.66, 361-375, 1959.
• Referans26 Stieglitz, M. and Tietz, H., "Matrixtransformationen von folgenraumen eine ergebnisüberischt", Math Z., Vol.154, 1-16, 1977.
• Referans27 Sulaiman, W.T., "On summability factors of infinite series", Proc. Amer. Math. Soc. Vol.115, 313-317, 1992.
• Referans28Wilansky, A., Summability Through Functional Analysis, North-Holland Mathematical Studies, vol. 85, Elsevier Science Publisher, 1984.