On Some New Rhaly Sequence Spaces and Rhaly Sections in BK-Space

Year 2024, Volume: 7 Issue: 4, 212 - 219

Abdulaziz Daş , Bilal Altay

https://doi.org/10.33434/cams.1514658

Abstract

In this paper, we introduce some new sequence spaces and sectional subspaces related to the Rhaly matrix and BK spaces. Furthermore, we investigate their relations and identities among these subspaces and duals.

Keywords

AK-space, Distinguished subspaces, Matrix domain, Rhaly matrix, rK-space

References

• J. Boos, Classical and Modern Methods in Summability, Oxford U. Press, Oxford, 2000.
• A. Wilansky, Summability Through Functional Analysis, North Holland, 1984.
• B. Altay, F. Basar, Certain topological properties and duals of the domain of a triangle matrix in a sequence space, J. Math. Anal. Appl., 336 (2007), 632–645.
• C. Aydın, F. Başar, Some new sequence spaces which include the spaces $l_p$ and $\ell_\infty$, Demonstr. Math., 38 (2005), 641–656.
• F. Başar, Summability Theory and its Applications, 2nd ed., CRC Press/Taylor & Francis Group, Boca Raton-London-New York, 2022.
• Ç . A. Bektaş, M. Et, R. Çolak, Generalized difference sequence spaces and their dual spaces, J. Math. Anal. Appl., 292 (2004), 423–432.
• M. C. Bişgin, A. Sönmez, Two new sequence spaces generated by the composition of m-th order generalized difference matrix and lambda matrix, J. Inequal. Appl., 2014, 274, (2014).
• M. Candan, Domain of the double sequential band matrix in the classical sequence spaces, J. Inequal. Appl., 2012, 281(2012).
• B. Choudhary, S. K. Mishra, A note on Köthe–Toeplitz duals of certain sequence spaces and their matrix transformations, Int. J. Math. Math. Sci., 18 (1995), 681–688.
• M. C. Dağlı, Two new fibonomial difference sequence spaces and related dual properties, Contrib. Math., 9(2024), 38-45.
• M. İlkhan, N. Şimşek, E. E. Kara, A new regular infinite matrix defined by Jordan totient function and its matrix domain in $\ell_p$, Math. Methods Appl. Sci., 44(9) (2021), 7622–7633.
• E. E. Kara, M. Bas¸arır, S. Konca, On some new weighted Euler sequence spaces and compact operators, Math. Inequal. Appl., 17(2) (2014), 649–664.
• E. E. Kara, M. İlkhan, On some Banach sequence spaces derived by a new band matrix, British J. Math. Comp. Sci., 9(2) (2015), 141–159.
• K. Kayaduman, M. Şengönül, The spaces of Cesaro almost convergent sequences and core theorems, Acta Math. Sci. Ser. B (Engl. Ed.), 32 (2012), 2265–2278.
• H. Kızmaz, On certain sequence spaces, Canad. Math. Bull., 24 (1981), 169–176.
• E. Malkowsky, V. Rakocevic, On matrix domains of triangles, Appl. Math. Comput., 189 (2007), 1146–1163.
• P. N. Ng, P. Y. Lee, Cesaro sequence spaces of non–absolute type, Comment. Math. Prace Mat., 20(2) (1978), 429–433.
• C. S. Wang, On Nörlund sequence space, Tamkang J. Math., 9 (1978), 269–274.
• T. Yaying, M. İ. Kara, B. Hazarika, E. E. Kara, A study on $q$-analogue of Catalan sequence spaces, Filomat 37(3)(2023),839- 850.
• K. Zeller, Allgemeine Eigenschaften von Limitierungsverfahren, Math. Z., 53 (1951), 463-487.
• K. Zeller,Abschnittskonvergenz in FK-Raumen, Math. Z., 55 (1951), 55-70.
• W.L.C. Sargent, On sectionally bounded BK-spaces, Math. Z., 83 (1964), 57-66.
• G. Bennett, Distinguished subspaces and summability invariants, Studia Math., 40 (1971), 225-234.
• A. O. Akdemir, M. T. Ersoy, H. Furkan, M. A. Ragusa, Some functional sections in topological sequence spaces, J. Funct. Spaces, 2022 (2022), Article ID 6449630, 7 pages, doi:10.1155/2022/6449630.
• M. Buntinas, Convergent and bounded Ces`aro sections in FK spaces, Math. Z., 121 (1971), 191-200.
• M. Buntinas, On Toeplitz sections in sequence spaces, Math. Proc. Cambridge Philos. Soc., 78 (1975), 451-460.
• I. Dağadur, On some subspaces of an FK-space, Math. Commun., 7 (2002), 15–20
• D.J. Fleming, Unconditional Toeplitz sections in sequence spaces, Math. Z., 194 (1987), 405-414.
• D. J. H. Garling, The b- and g-duality of sequence spaces, Math. Proc. Cambridge Philos. Soc., 63 (1967), 963-981.
• D. J. H. Garling, On topological sequence spaces, Math. Proc. Cambridge Philos. Soc., 63 (1967), 997-1019.
• G. Goes, S. Goes, Sequences of bounded variation and sequences of Fourier coefficients I, Math. Z., 118 (1970), 93-102.
• G. Goes, Summan von FK-R˘aumen funktionale Abschnıttskonvergenz und Umkehrsatz, Tohoku Math. J., 26 (1974), 487–504.
• G. Meyers, On Toeplitz sections in FK-spaces, Studia Math., 51 (1974), 23-33.
• J. J. Sember, On unconditional section boundedness in sequence spaces, Rocky Mountain J. Math., 7 (1977), 699-706.
• H. G. İnce, Combinations of distinguished subsets and Cesa ro conullity, Commun. Math. Anal., 1(2) (2006), 91-99.
• M. T. Ersoy, B. Altay, H. Furkan, On Riesz sections in sequence spaces, J. Adv. Math. Comp. Sci., 24(3) (2017), 1-10.
• T. Bilgin, M. Karakus¸, On some distinguished subspaces and relationship between duals, J. Adv. Math. Math. Edu., 1(1) (2018), 5-15.
• M. Karakuş, T. Bilgin, On some new FK spaces obtained from summability matrix, J. Universal Math., 3(1) (2020), 66-76.
• G. Leibowitz, Rhaly Matrices, J.Math. Anal. Appl., 128(1) (1987), 272–286.
• Rhaly, H. C., Terraced matrices, Bull. London Math. Soc., 21(4) (1989), 399–406.
• M. Stieglitz, H. Tietz, Matrix transformationen von folgenraumen eine ergebnis¨ubersict, Math. Z., 154 (1977), 1–16.