Home Linear isomorphic spaces of Cesàro–Nörlund operator, their duals and matrix transformations
Article
Licensed
Unlicensed Requires Authentication

Linear isomorphic spaces of Cesàro–Nörlund operator, their duals and matrix transformations

  • Uday Pratap Singh , Swati Jasrotia and Kuldip Raj EMAIL logo
Published/Copyright: November 24, 2022

Abstract

In this article, we introduce and study sequence spaces of Cesàro–Nörlund operators of order n associated with a sequence of Orlicz functions. We obtain some topological properties and Schauder basis of these sequence spaces. Moreover, we compute the α-, β- and γ-duals and the matrix transformations of these newly formed sequence spaces. Finally, we prove that these sequence spaces are of Banach–Saks type p and have a weak fixed-point property.

MSC 2010: 40C05; 46A45

References

[1] F. Başar, Summability Theory and its Applications, Bentham Science, Oak Park, 2012. 10.2174/97816080545231120101Search in Google Scholar

[2] B. Beauzamy, Banach–Saks properties and spreading models, Math. Scand. 44 (1979), no. 2, 357–384. 10.7146/math.scand.a-11818Search in Google Scholar

[3] A. J. Dutta, A. Esi and B. C. Tripathy, Statistically convergent triple sequence spaces defined by Orlicz function, J. Math. Anal. 4 (2013), no. 2, 16–22. Search in Google Scholar

[4] J. García Falset, The fixed point property in Banach spaces with the NUS-property, J. Math. Anal. Appl. 215 (1997), no. 2, 532–542. 10.1006/jmaa.1997.5657Search in Google Scholar

[5] G. H. Hardy, An inequality for Hausdorff means, J. London Math. Soc. 18 (1943), 46–50. 10.1112/jlms/s1-18.1.46Search in Google Scholar

[6] G. H. Hardy, Divergent Series, Oxford University, Oxford, 1949. Search in Google Scholar

[7] G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, 2nd ed., Cambridge University, Cambridge, 1952. Search in Google Scholar

[8] H. Knaust, Orlicz sequence spaces of Banach–Saks type, Arch. Math. (Basel) 59 (1992), no. 6, 562–565. 10.1007/BF01194848Search in Google Scholar

[9] G. Köthe and O. Toeplitz, Lineare Räume mit unendlich vielen Koordinaten und Ringe unendlicher Matrizen, J. Reine Angew. Math. 171 (1934), 193–226. 10.1515/crll.1934.171.193Search in Google Scholar

[10] J. Lindenstrauss and L. Tzafriri, On Orlicz sequence spaces, Israel J. Math. 10 (1971), 379–390. 10.1007/BF02771656Search in Google Scholar

[11] F. M. Mears, The inverse Nörlund mean, Ann. of Math. (2) 44 (1943), 401–410. 10.2307/1968971Search in Google Scholar

[12] M. Mursaleen, Applied summability methods, Springer Briefs Math., Springer, Cham, 2014. 10.1007/978-3-319-04609-9Search in Google Scholar

[13] J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Math. 1034, Springer, Berlin, 2006. Search in Google Scholar

[14] K. Raj and A. Kılıçman, On certain generalized paranormed spaces, J. Inequal. Appl. 2015 (2015), Paper No. 37. 10.1186/s13660-015-0565-zSearch in Google Scholar

[15] H. Roopaei, D. Foroutannia, M. İlkhan and E. E. Kara, Cesàro spaces and norm of operators on these matrix domains, Mediterr. J. Math. 17 (2020), no. 4, Paper No. 121. 10.1007/s00009-020-01557-9Search in Google Scholar

[16] M. Stieglitz and H. Tietz, Matrixtransformationen von Folgenräumen. Eine Ergebnisübersicht, Math. Z. 154 (1977), no. 1, 1–16. 10.1007/BF01215107Search in Google Scholar

[17] B. C. Tripathy, A. Esi and B. Tripathy, On a new type of generalized difference Cesàro sequence spaces, Soochow J. Math. 31 (2005), no. 3, 333–340. Search in Google Scholar

[18] C. S. Wang, On Nörlund sequence spaces, Tamkang J. Math. 9 (1978), no. 2, 269–274. Search in Google Scholar

[19] A. Wilansky, Summability Through Functional Analysis, North-Holland, Amsterdam, 1984. Search in Google Scholar

[20] M. Yeşilkayagil and F. Başar, On the paranormed Nörlund sequence space of nonabsolute type, Abstr. Appl. Anal. 2014 (2014), Article ID 858704. 10.1155/2014/858704Search in Google Scholar

[21] M. Yeşilkayagil and F. Başar, Domain of the Nörlund matrix in some of Maddox’s spaces, Proc. Nat. Acad. Sci. India Sect. A 87 (2017), no. 3, 363–371. 10.1007/s40010-017-0359-4Search in Google Scholar

[22] M. Zeltser, M. Mursaleen and S. A. Mohiuddine, On almost conservative matrix methods for double sequence spaces, Publ. Math. Debrecen 75 (2009), no. 3–4, 387–399. 10.5486/PMD.2009.4396Search in Google Scholar

Received: 2021-04-15
Revised: 2021-09-17
Accepted: 2021-11-01
Published Online: 2022-11-24
Published in Print: 2023-12-01

© 2022 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 23.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/jaa-2022-2008/html
Scroll to top button