Korean J. Math. Vol. 28 No. 2 (2020) pp.205-221
DOI: https://doi.org/10.11568/kjm.2020.28.2.205

A new paranormed series space using Euler totient means and some matrix transformations

Main Article Content

G. Canan Hazar Güleç
Merve İlkhan


Paranormed spaces are important as a generalization of the normed spaces in terms of having more general properties. The aim of this study is to introduce a new paranormed space $ \left\vert \phi _{z}\right\vert \left( p\right) $ over the paranormed space $ \ell \left( p\right) $ using Euler totient means, where $p=\left( p_{k}\right) $ is a bounded sequence of positive real numbers. Besides this, we investigate topological properties and compute the $ \alpha -,\beta -,$ and $\gamma $ duals of this paranormed space. Finally, we characterize the classes of infinite matrices $(\left\vert \phi_{z}\right\vert \left( p\right) ,\lambda )$ and $(\lambda ,\left\vert \phi_{z}\right\vert \left( p\right) ),$\ where $\lambda $ is any given sequence space.

Article Details


[1] B. Altay and F. Ba ̧sar, On the paranormed Riesz sequence spaces of non-absolute type, Southeast Asian Bull. Math. 26 (5) (2003), 701–715. Google Scholar

[2] B. Altay and F. Ba ̧sar, Generalization of the sequence spaces l(p) derived by weighted mean, J. Math. Anal. Appl. 330 (2007), 174–185. Google Scholar

[3] B. Altay, F. Ba ̧sar and M. Mursaleen, On the Euler sequence spaces which include the spaces l (p) and l∞ I, Inform. Sci. 176 (10) (2006), 1450–1462. Google Scholar

[4] C. Aydın and F. Ba ̧sar, Some new sequence spaces which incule the spaces lp and l∞, Demonstratio Math. 38(3) (2005), 641-656. Google Scholar

[5] C. Aydın and F. Ba ̧sar, Some generalizations of the sequence space, Iran. J. Sci. Technol. Trans. A Sci. 30 (A2) (2006), 175–190. Google Scholar

[6] B. Altay, F. Ba ̧sar and M. Mursaleen, Some generalizations of the space bvp of p-bounded variation sequences, Nonlinear Anal. 68(2) (2008), 273–287. Google Scholar

[7] B. Choudhary and S. K. Mishra, On K ̈othe-Toeplitz duals of certain sequence spaces and their matrix transformations, Indian J. Pure Appl. Math. 24 (1993), 291–301. Google Scholar

[8] S. Demiriz and E. E. Kara, Some new paranormed difference sequence spaces derived by Fibonacci numbers, Miskolc Math. Notes 16 (2) (2015), 907–923. Google Scholar

[9] K. G. Grosse-Erdmann, Matrix transformations between the sequence spaces of Maddox, J. Math. Anal. Appl. 180 (1993), 223–238. Google Scholar

[10] F. G ̈ok ̧ce and M. A. Sarıg ̈ol, A new series space and matrix operators with applications, Kuwait J. Sci. 45 (4) (2018), 1–8. Google Scholar

[11] G. C. Hazar and M. A. Sarıg ̈ol, Absolute Ces`aro series spaces and matrix operators, Acta App. Math. 154 (2018), 153–165. Google Scholar

[12] G. C. Hazar Gu ̈le ̧c, Compact matrix operators on absolute Ces`aro spaces, Numer. Funct. Anal. Optim. 41 (1) (2020), 1–15. Google Scholar

[13] M. I ̇lkhan and E. E. Kara, A new Banach space defined by Euler totient matrix operator, Oper. Matrices 13 (2) (2019), 527–544. Google Scholar

[14] M. I ̇lkhan, S. Demiriz and and E. E. Kara, A new paranormed sequence space defined by Euler totient matrix, Karaelmas Sci. Eng. J. 9 (2) (2019), 277–282. Google Scholar

[15] M. I ̇lkhan and G. C. Hazar Gu ̈le ̧c, A study on absolute Euler totient series space and certain matrix transformations, Mugla J. Sci. Technol. 6 (1) (2020), 112–119. Google Scholar

[16] E.E.Kara,M.O ̈ztu ̈rkandM.Ba ̧sarır, Some topological and geometric properties of generalized Euler sequence spaces, Math. Slovaca 60 (3) (2010), 385–398. Google Scholar

[17] E. E. Kara, Some topological and geometrical properties of new Banach sequence spaces, J. Inequal. Appl. 2013 (2013), 38. Google Scholar

[18] E. E. Kara and M. I ̇lkhan, Some properties of generalized Fibonacci sequence spaces, Linear Multilinear Algebra 64(11) (2016), 2208–2223. Google Scholar

[19] C. G. Lascarides and I. J. Maddox, Matrix transformations between some classes of sequences, Proc. Camb. Phil. Soc. 68 (1970), 99–104. Google Scholar

[20] I. J., Maddox, Paranormed sequence spaces generated by infinite matrices, Proc. Cambridge Philos. Soc. 64 (1968), 335–340. Google Scholar

[21] I. J. Maddox, Spaces of strongly summable sequences, Quart. J. Math. Oxford 18 (2) (1967), 345–355. Google Scholar

[22] E. Malkowsky, V. RakoVcevi c and S. ZVivkovi c, Matrix transformations between the sequence space bvk and certain BK spaces, Bull. Cl. Sci. Math. Nat. Sci. Math. 123 (27) (2002), 33-46. Google Scholar

[23] E. Malkowsky and E. Sava ̧s, Matrix transformations between sequence spaces of generalized weighted mean, Appl. Math. Comput. 147 (2004), 333–345. Google Scholar

[24] M. Mursaleen, F. Ba ̧sar and B. Altay, On the Euler sequence spaces which include the spaces lp and l∞ II, Nonlinear Anal. 65 (3) (2006), 707–717. Google Scholar

[25] H. Nakano, Modulared sequence spaces, Proc. Jpn. Acad. 27 (2) (1951), 508–512. Google Scholar

[26] P.-N, Ng, and P.-Y., Lee, Ces`aro sequence spaces of non-absolute type, Comment. Math. Prace Mat. 20 (2) (1978), 429–433. Google Scholar

[27] M. A. Sarıg ̈ol, Spaces of series summable by absolute Ces `aro and matrix operators, Comm. Math Appl. 7 (1) (2016), 11–22. Google Scholar

[28] M. A. Sarıg ̈ol, On the local properties of factored Fourier series, Appl. Math. Comp. 216 (2010), 3386–3390. Google Scholar

[29] S. Simons, The sequence spaces l(pv) and m(pv), Proc. London Math. Soc. 15 (3) (1965), 422–436. Google Scholar

[30] I. Schoenberg, I., The integrability of certain functions and related summability methods, Amer. Math. Monthly 66 (1959), 361–375. Google Scholar

[31] A. Wilansky, Summability Through Functional Analysis, North-Holland Mathematical Studies, Elsevier Science Publisher (1984). Google Scholar

[32] M. Ye ̧silkayagil and F. Ba ̧sar, On the paranormed N ̈orlund sequence space of nonabsolute type, Abstr. Appl. Anal. 2014 (2014), Article ID: 858704. Google Scholar

[33] P. Zengin Alp and M. I ̇lkhan, On the difference sequence space lp(Tˆq), Math. Sci. Appl. E-Notes 7 (2) (2019), 161–173. Google Scholar