A New Paranormed Series Space and Matrix Transformations

Year 2020, Volume: 8 Issue: 1, 91 - 99, 20.03.2020

G. Canan Hazar Güleç

https://doi.org/10.36753/mathenot.627066

Abstract

Recently, Hazar and Sarıgöl have defined and studied the series space |C₋₁|_{p} for 1≤p<∞ in [1]. The aim of this study is to introduce a new paranormed space |C₋₁|(p), where p=(p_{k}) is a bounded sequence of positive real numbers, which extends the results of Hazar and Sarıgöl in [1] to paranormed space. Besides this, we investigate topological properties and compute the α-,β-, and γ duals of this paranormed space. Finally, we characterize the classes of infinite matrices (|C₋₁|(p),μ) and (μ,|C₋₁|(p)), where μ is any given sequence spaces

Keywords

Paranormed sequence spaces, Absolute summability, Cesàro means; Matrix transformations

References

• 1 : Altay, B. and Başar, F., On the paranormed Riesz sequence spaces of non-absolute type, Southeast Asian Bull. Math. 26(5)(2003), 701-715. 2 : Altay, B. and Başar, F., Some paranormed Riesz sequence spaces of non-absolute type, Southea st Asian Bull. Math. 30 (4)(2006), 591-608 3 : Aydın, C. and Başar, F., Some generalizations of the sequence space, Iranian Journal of Science and Technology, Transaction A: Science. 30, No.A2 (2006). 4 : Başar, F. and Altay, B., Matrix mappings on the space bs(p) and its α-,β-,γ- duals, Aligarh Bull. Math.21(1) (2002), 79-91. 5 : Başar, F., Altay, B. and Mursaleen, M., Some generalizations of the space bv_{p} of p-bounded variation sequences, Nonlinear Analysis: Theory, Methods & Applications, 68 (2) (2008) 273-287. 6 : Flett, T.M., On an extension of absolute summability and some theorems of Littlewood and Paley, Proc. London Math. Soc. 7 (1957), 113-141. 7 : Hardy, G. H., Divergent Series, Oxford, 1949. 8 : Grosse-Erdmann, K.G., Matrix transformations between the sequence spaces of Maddox, J. Math. Anal. Appl., 180, (1993), 223-238. 9 : Gokce, F., and Sarıgöl M.A., A new series space |N_{p}^{θ}|(μ) and matrix operators with applications, Kuwait Journal of Science, 45 (4), (2018), 1-8. 10 : Hazar Güleç, G.C. and Sarıgöl M. A., Compact and Matrix Operators on the Space |C,-1|_{k}, J. Comput. Anal. Appl., 25(6), (2018), 1014-1024. 11 : Hazar, G. C. and Sarıgöl M. A., "Absolute Cesàro series spaces and matrix operators", Acta App. Math., 154, 153--165 (2018). 12 : Hazar Güleç, G. C., Compact Matrix Operators on Absolute Cesàro Spaces, Numer. Funct. Anal. Optim., DOI: 10.1080/01630563.2019.1633665 13 : Kara, E. E., and Demiriz, S., Some New Paranormed Difference Sequence Spaces Derived by Fibonacci Numbers, Miskolc Mathematical Notes, 16(2) (2015), 907-923. 14 : Kara, E. E. and İlkhan, M., Some properties of generalized Fibonacci sequence spaces, Linear and Multilinear Algebra, 64 (11) (2016), 2208-2223. 15 : Lascarides, C.G. and Maddox, I.J., Matrix transformations between some classes of sequences, Proc. Camb. Phil. Soc. 68, (1970), 99-104. 16 : Maddox, I.J., Paranormed sequence spaces generated by infinite matrices, Proc. Cambridge Philos. Soc. 64, (1968), 335-340. 17 : Maddox, I.J., Spaces of strongly summable sequences, Quart. J. Math. Oxford 18(2) (1967), 345-355. 18 : Maji, A. and Srivastava, P., Some Paranormed Difference Sequence Spaces of Order m Derived by Generalized Means and Compact Operators.,(2013),arXiv:1308.2667v2 19 : Malkowsky, E., Recent results in the theory of matrix transformations in sequence spaces, Mat. Vesnik 49 (1997), 187-196 20 : Nakano, H., Modulared sequence spaces, Proc. Jpn. Acad. 27(2) (1951), 508-512. 21 : Yeşilkayagil, M., and Başar, F., On the paranormed Nörlund sequence space of nonabsolute type, Abstract and Applied Analysis, Vol. 2014, Article ID: 858704. 22 : Sarıgöl, M.A., Spaces of Series Summable by Absolute Cesàro and Matrix Operators, Comm. Math Appl. 7 (1) (2016) 11-22. 23 : Simons, S., The sequence spaces ℓ(p_{v}) and m(p_{v}), Proc. London Math. Soc. 15(3) (1965), 422-436. 24 : Thorpe, B., Matrix transformations of Cesàro summable Series, Acta Math. Hung., 48(3-4), (1986), 255-265 25 : Wilansky, A., Summability Through Functional Analysis, North-Holland Mathematical Studies, vol. 85, Elsevier Science Publisher, 1984.