On the paranormed binomial sequence spaces

Yıl 2018, Cilt: 1 Sayı: 3, 137 - 147, 30.09.2018

Hacer Bilgin Ellidokuzoğlu , Serkan Demiriz , Ali Köseoğlu

https://doi.org/10.32323/ujma.395247

Cited By: 3

Öz

In this paper the sequence spaces $b_0^{r,s}(p)$, $b_c^{r,s}(p)$, $b_{\infty}^{r,s}(p)$ and $b^{r,s}(p)$ which are the generalization of the classical Maddox's paranormed sequence spaces have been introduced and proved that the spaces $b_0^{r,s}(p)$, $b_c^{r,s}(p)$, $b_{\infty}^{r,s}(p)$ and $b^{r,s}(p)$ are linearly isomorphic to spaces $c_0(p)$, $c(p)$, $\ell_{\infty}(p)$ and $\ell(p)$, respectively. Besides this, the $\alpha-,\beta-$ and $\gamma-$duals of the spaces $b_0^{r,s}(p)$, $b_c^{r,s}(p)$, and $b^{r,s}(p)$ have been computed, their bases have been constructed and some topological properties of these spaces have been studied. Finally, the classes of matrices $(b_0^{r,s}(p) : \mu)$, $(b^{r,s}_c(p): \mu)$ and $(b^{r,s}(p): \mu)$ have been characterized, where $\mu$ is one of the sequence spaces $\ell_\infty,c$ and $c_0$ and derives the other characterizations for the special cases of $\mu$.

Anahtar Kelimeler

Binomial sequence spaces, Paranorm, Matrix domain, Matrix transformations

Kaynakça

• [1] B. Altay, F. Başar, On the paranormed Riesz sequence spaces of non-absolute type, Southeast Asian Bull. Math., 26, 701-715 (2002).
• [2] B. Altay, F. Başar, Some paranormed Riesz sequence spaces of non-absolute type, Southeast Asian Bull. Math., 30, 591-608 (2006).
• [3] F. Başar, B. Altay, Matrix mappings on the space bs(p) and its a−,b− and g−duals, Aligarh Bull. Math., 21(1), 79-91 (2002).
• [4] F. Başar, Infinite matrices and almost boundedness, Boll. Un. Mat. Ital., 6(7), 395-402 (1992).
• [5] M. C. Bişgin, The binomial sequence spaces of nonabsolute type, J. Inequal. Appl. 309 (2016).
• [6] M. C. Bişgin, The binomial sequence spaces which include the spaces ℓp and ℓ¥ and geometric properties, J. Inequal. Appl. 304 (2016).
• [7] B. Choudhary, S. K. Mishra, On Köthe-Toeplitz duals of certain sequence spaces and their matrix transformations, Indian J. Pure Appl. Math., 24(5), 291-301 (1993).
• [8] S. Demiriz, C. C¸ akan, On Some New Paranormed Euler Sequence Spaces and Euler Core, Acta Math. Sin.(Eng. Ser.), 26(7), 1207-1222 (2010).
• [9] S. Demiriz, H. B. Ellidokuzoğlu, On The Paranormed Taylor Sequence Spaces, Konuralp Journal Of Mathematics, 4(2), 132-148 (2016).
• [10] K. G. Grosse-Erdmann, Matrix transformations between the sequence spaces of Maddox. J. Math. Anal. Appl., 180, 223-238 (1993).
• [11] A. Jarrah and E. Malkowsky, BK spaces, bases and linear operators, Rend. Circ. Mat. Palermo, 52(2), 177-191 (1990).
• [12] E. E. Kara and M. lkhan, On some Banach sequence spaces derived by a new band matrix, Br. J. Math. Comput. Sci., 9(2), 141-159 (2015). [13] E. E. Kara and M. Ilkhan, Some properties of generalized Fibonacci sequence spaces, Linear Multilinear Algebra, 64(11), 2208-2223 (2016). [14] M. Kirişci, On the Taylor sequence spaces of nonabsulate type which include the spaces c0 and c, J. Math. Anal., 6(2), 22-35 (2015).
• [15] M. Kirişci, The application domain of infinite matrices with algorithms, Univ. J. Math. Appl., 1(1), 1-9 (2018).
• [16] M. Candan and A. Güneş, Paranormed sequence space of non-absolute type founded using generalized difference matrix, Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci. 85(2), 269-276 (2015).
• [17] C. G. Lascarides and I. J. Maddox, Matrix transformations between some classes of sequences, Proc.Camb. Phil. Soc., 68, 99-104 (1970).
• [18] I.J. Maddox, Elements of Functional Analysis, second ed., The University Press, Cambridge, 1988.
• [19] I. J. Maddox, Paranormed sequence spaces generated by infinite matrices, Proc. Camb. Phios. Soc., 64, 335-340 (1968).
• [20] H. Nakano, Modulared sequence spaces, Proc. Jpn. Acad., 27(2), 508-512 (1951).
• [21] S. Simons, The sequence spaces ℓ(pv) and m(pv). Proc. London Math. Soc., 15(3), 422-436 (1965).