A new perspective on paranormed Riesz sequence space of non-absolute type
DOI:
https://doi.org/10.14419/gjma.v3i4.5573Keywords:
Paranormed sequence space, alpha-, beta- and gamma-duals and matrix mappings.Abstract
The current article mainly dwells on introducing Riesz sequence space \(r^{q}(\widetilde{B}_{u}^{p})\) which generalized the prior studies of Candan and Güneş [28], Candan and Kılınç [30]  and consists of all sequences whose \(R_{u}^{q}\widetilde{B}\)-transforms are in the space \(\ell(p)\), where \(\widetilde{B}=B(r_{n},s_{n})\) stands for double sequential band matrix \((r_{n})^{\infty}_{n=0}\) and \((s_{n})^{\infty}_{n=0}\) are given convergent sequences of positive real numbers. Some topological properties of the new brand sequence space have been investigated as well as \(\alpha\)- \(\beta\)-and \(\gamma\)-duals. Additionally, we have also constructed the basis of \(r^{q}(\widetilde{B}_{u}^{p})\). Eventually, we characterize a matrix class on the sequence space. These results are more general and more comprehensive than the corresponding results in the literature.
References
[1] F. BaÅŸar, Summability Theory and Its Applications, Bentham Science Publishers, e-books, Monographs, xi+405 pp., Ä°stanbul 2012, ISB:978-1-60805-252-3.
[2] I. J. Maddox, Paranormed sequence spaces generated by infinite matrices, Proc. Cambridge. Philos. Soc. (64)(1968), 335--340.
[3] I. J. Maddox, Spaces of strongly summable sequences, Quart. J. Math. Oxford, (18) (2) (1967), 345--355.
[4] S. Simons, The sequence spaces and , Proc. London Math. Soc., (15) (3) (1965), 422--436.
[5] H. Nakano, Modulared sequence spaces, Proc. Japan Acad. (27) (2)(1951), 508--512.
[6] P.N. Ng and P.Y. Lee, Cesà ro sequence spaces of non-absolute type, Comment. Math. Prace Mat. (20) (1978), no.2, 429-433.
[7] B. Altay and F. BaÅŸar, On the paranormed Riesz sequence spaces of non-absolute type, Southeast Asian Bull. Math. (26) (2002), 701--715.
[8] B. Altay, F. BaÅŸar, M. Mursaleen, On the Euler sequence spaces which include the spaces and I, Inform. Sci. (176) (10)(2006), 1450--1462.
[9] M. Mursaleen, F. BaÅŸar, B. Altay, On the Euler sequence spaces which include the spaces and II, Nonlinear Anal. (65) (3)(2006), 707--717.
[10] E.E. Kara, M. Öztürk, M. Başarır, Some topological and geometric properties of generalized Euler sequence spaces, Math. Slovaca (60) (3)(2010), 385--398.
[11] E. Malkowsky and E. SavaÅŸ, Matrix transformations between sequence spaces of generalized weighted means, Appl. Math. Comput. (147) (2004), 333--345.
[12] B. Altay, F. BaÅŸar, Generalization of the sequence space derived by weighted mean, J. Math. Anal. Appl. (330) (2007), 174--185.
[13] C. Aydın and F. Başar, Some new sequence spaces which include the spaces and , Demonstratio Math. (38) (3)(2005), 641-656.
[14] C. Aydın and F. Başar, Some generalizations of the sequence spaces , Iran. J. Sci. Technol. Trans.A.Sci. (30) (A2)(2006), 175--190.
[15] E. Malkowsky, V. Rakocevic, S. Zivkovic, Matrix transformations between the sequence space and certain spaces, Bull. Cl. Sci. Math. Nat. Sci. Math. (27) (2002), 33-46.
[16] F. BaÅŸar and B. Altay, On the space of sequences of -bounded variation and related matrix mappings, Ukranian Math. J. (55) (2003), 136-147.
[17] B. Altay and F. BaÅŸar, Some paranormed sequence spaces of non-absolute type derived by weighted mean, J. Math. Anal. Appl. (319) (2)(2006), 494--508.
[18] B. Choudhary and S. K. Mishra, On Köthe-Toeplitz duals of certain sequence spaces and their matrix transformations, Indian J. Pure Appl. Math. (24) (1993), no.5, 291--301.
[19] M. Mursaleen, A.K. Noman, On some new sequence spaces of non-absolute type related to the spaces and I, Filomat (25) (2)(2011), 33--51.
[20] M. Kirişçi, F.Başar, Some new sequence spaces derived by the domain of generalized difference matrix, Comput. Math. Appl. (60) (5)(2010), 1299--1309.
[21] M. Candan, Domain of the double sequential band matrix in the classical sequence spaces, J. Inequal. Appl. (2012), 2012:281, 15 pp.
[22] F. Başar, M. Kirişçi, Almost convergence and generalized difference matrix, Comput. Math. Appl. (61) (3)(2011), 602--611.
[23] M. Candan, Almost convergence and double sequential band matrix, Acta. Math. Sci. (34B) (2)(2014), 354--366.
[24] M. Candan, A new sequence space isomorphic to the space and compact operators, J. Math. Comput. Sci. 4 (2014), No.2, 306-334.
[25] M. Candan, Domain of the double sequential band matrix in the spaces of convergent and null sequences, Adv. Difference Edu.(2014), 2014:163, 18 pp.
[26] M. Candan, Some new sequence spaces derived from the spaces of bounded, convergent and null sequences, Int. J. Mod. Math. Sci.,(12) (2) (2014), 74-87.
[27] M. Candan, A new approach on the spaces of generalized Fibonacci difference null and convergent sequences, Math. AEterna. (1) (5) (2015), 191-210.
[28] M. Candan and A. Güneş, Paranormed sequence space of non-absolute type founded using generalized difference matrix, Proc. Nat. Acad. Sci. India Sect. A., (85) (2) (2015), 269-276.
[29] M. Candan and K. Kayaduman, Almost convergent sequence space derived by generalized Fibonacci matrix and Fibonacci core, Brithish J. Math. Comput. Sci., (7) (2) (2015), 150-167.
[30] M. Candan and G. Kılınç, A different look for paranormed Riesz sequence space of derived by Fibonacci Matrix, Konuralp J. Math. Vol.3, No.2, (2015), 62-76.
[31] M. Candan, E.E. Kara, A study on topological and geometrical characteristics of new Banach sequence spaces, Gulf J. Math. Vol.3, No.4, (2015), 67-84.
[32] M. Kirişçi, On the spaces of Euler almost null and Euler almost convergent sequences, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 62(1), 1-16 (2013).
[33] M. Kirişçi, Almost convergence and generalized weighted mean, AIP Conf. Proc., vol. 1470, pp. 191-194 (2012).
[34] M. Kirişçi, Almost Convergence and Generalized Weighted Mean II, J. Inequal. Appl. (2014),2014:93, 13 pp.
[35] B. Altay and F. BaÅŸar, Some paranormed sequence spaces derived by generalized weighted mean, J. Math. Anal. Appl. (319)(2006), 494--508.
[36] H. Polat, V. Karakaya, N. ÅžimÅŸek, Difference sequence spaces derived by generalized weighted mean, Appl. Math. Lett. (24) (5)(2011), 608--314.
[37] M. Başarır, On the generalized Riesz -difference sequence spaces, Filomat (24) (4)(2010), 35--52.
[38] B. Altay, F. BaÅŸar, On the fine spectrum of the generalized difference operator over the sequence and , Int. J. Math. Sci. (18) (2008), 3005--3013.
[39] S. Demiriz, C. Çakan, Some topolojical and geometrical properties of a new difference sequence space, Abstr. Appl. Anal. doi:10.1155/2011/213878, 14 pp.
[40] M. Başarır, M. Öztürk, On the Riesz diference sequence space, Rend. Circ. Mat. Palermo (57) (2008), 377-389.
[41] M. Başarır, Paranormed Cesà ro difference sequence space and related matrix transformation, Doğa Tr. J. Math. (15) (1991), 14--19.
[42] M. Et, M. Işık, On pa-dual spaces of generalized difference sequence spaces, Appl. Math. Lett. (25) (10)(2012), 1486--1489.
[43] M. Et, Generalized Cesà ro difference sequence spaces of non-absolute type involving lacunary sequence spaces, Appl. Math. Comput. (219) (17)(2013), 9372--9376.
[44] M. Başarır, E.E. Kara, On the order difference sequence space of generalized weighted mean and compact operator, Acta. Math. Sci. (33B) (3)(2013), 1--18.
[45] M. Başarır, E.E. Kara, On the difference sequence space derived by generalized weighted mean and compact operators, J. Math. Anal. Appl. (391) (2012), 67--81.
[46] M. Başarır, M. Kayıkçı, On the generalized Riesz difference sequence space and beta-property J. Inequal. Appl. (2009), ID 385029, 18pp.
[47] M. Başarır, E.E. Kara, On some difference sequence spaces of weighted means and compact operators, Ann. Funct. Anal. (2) (2)(2011), 116--131.
[48] M. Başarır, E.E. Kara, On compact operators on the Riesz difference sequence space, Iran. J. Sci. Technol. Trans.(35A) (4)(2011), 279--285.
[49] M. Başarır, E.E. Kara, On compact operators on the Riesz difference sequence space-II, Iran. J. Sci. Technol. Trans. (36A) (3)(2012), 371--376.
[50] M. Başarır, M. Öztürk, On some Generalized difference Riesz Sequence Spaces and Uniform Opial Property, J. Inequal. Appl. (2011), 2011:485730, 17 pp.
[51] Ş. Konca, M. Başarır, Generalized difference sequence spaces associated with a multiplier sequence on a real normed space, J. Inequal. Appl. (2013), 2013:335, 12 pp.
[52] Ş. Konca, M. Başarır, On some spaces of almost lacunary convergent sequences derived by Riesz mean and weighted almost lacunary statistical convergence in a real normedspace, J. Inequal. Appl. (2014), 2013:81, 11 pp.
[53] M. Candan, Ä°. Solak, On some Difference Sequence Spaces Generated by Infinite Matrices, International Journal of Pure and Applied Mathematics, V.25(1), 79-85, 2005.
[54] M. Candan, Ä°. Solak, On New Difference Sequence Spaces Generated
by Infinite Matrices, International Journal of Science and Tecnology, V 1(1), 15-17, 2006.
[55] M. Candan, Some new sequence spaces defined by a modulus function and an
infinite matrix in a seminormed space, J. Math. Anal., 3(2)(2012), 1-9.
[56] Y. Yılmaz, M.K. Özdemir, İ. Solak, M. Candan, Operators on some vector-valued Orlicz sequence spaces, F.Ü. Fen ve Mühendislik Dergisi, 17(1), 59-71, (2005).
[57] H. Kızmaz, On certain sequence spaces}, Canad. Math. Bull. (24) (2)(1981), 169--176.
[58] B. Altay, F. BaÅŸar, The matrix domain and the fine spectrum of the difference operator on the sequence space , , Commun. Math. Anal., Commun. Math. Anal. (2) (2)(2007), 1--11.
[59] F. BaÅŸar, B. Altay, On the space of sequences of bounded variation and related matrix mappings, Ukrainian Math. J. (55) (1)(2003), 136--147.
[60] R. Çolak, M. Et, E. Malkowsky, Some Topics of Sequence Spaces, Lecture Notes in Mathematics, Fırat Univ. Elazığ, Turkey, 2004, pp. 1--63, Fırat Univ. Press, 2004, ISBN: 975-394-038-6.
[61] R. Çolak, M. Et, On some generalized difference sequence spaces and related matrix transformations, Hokkaido Math. J.,(26) (3) (1997), 483--492.
[62] E. Malkowsky, S.D. Parashar, Matrix transformations in space of bounded and convergent difference sequence of order , Analysis (17) (1997), 87--97.
[63] H. Polat, F. BaÅŸar, Some Euler spaces of difference sequences of order , Acta Math. Sci. (27B) (2)(2007), 254--266.
[64] B. Altay, On the space of summable difference sequences of order , , Stud. Sci. Math. Hungar. (43) (4)(2006), 387--402.
[65] N.A. Sheikh, A.H. Ganie, A new paranormed sequence space and some matrix transformations, Acta Math. Acad. Paedago. Nyregy.,(28) (2012), 47-58.
[66] A.H. Ganie, N.A. Sheikh, New type of paranormed sequence space of non-absolute type and a matrix transformation, Int. J. Of Mod. Math. Sci, (8) (2) (2013), 196-211.
[67] K. -G. Grosse-Erdmann, Matrix transformations between the sequence spaces of Maddox, J. Math. Anal. Appl. (180) (1993), 223--238.
[68] C.G. Lascarides, I.J. Maddox, Matrix transformations between some classes of sequences, Proc. Cambridge Philos.Soc. (68) (1970), 99--104.
Downloads
Published
Issue
Section
License
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal''s published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).