Distinguished subspaces in topological sequence spaces theory

Merve Temizer Ersoy; Hasan Furkan; Merve Temizer Ersoy; Hasan Furkan

doi:10.3934/math.2020183

AIMS Mathematics

2020, Volume 5, Issue 4: 2858-2868. doi: 10.3934/math.2020183

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Research article

Distinguished subspaces in topological sequence spaces theory

Merve Temizer Ersoy ^,,
Hasan Furkan

Department of Mathematics, Kahramanmaraş Sütçü İmam University, Kahramanmaraş-46100, Turkey

Received: 23 October 2019 Accepted: 16 March 2020 Published: 18 March 2020
MSC : 46A45, 46A20

In this paper, we study $R_{\lambda}$-semiconservative $FK$-spaces for Riesz-method defined by the Riesz matrix $(R)$ and give some characterizations. We show that if $\ell_{A}$ is $\ell$-replaceable, then $A$ can not be $R_{\lambda}$-semiconservative and also if $X_{A}$ is $R_{\lambda}$-conull $FK$-space then it must be $R_{\lambda}$-semiconservative space. In addition, we determine a new $r(\lambda)$ and $rb(\lambda)$ type duality of a sequence space $X$ containing $\varphi$. The paper aims to develop some new subspaces which each one has its own value on topological sequence spaces theory. These subspaces are called as $R_{\lambda}S; R_{\lambda}W; R_{\lambda}F^{+}; $ and $R_{\lambda}B^{+}$ for a locally convex $FK$-space X containing $\varphi$. The subspaces mentioned in the work requires some serious studies and they arose independently from the literature which was done at the recent stage of the development of summability through functional analysis.
- topological sequence space,
- multiplier spaces,
- semiconcervative FK-spaces
Citation: Merve Temizer Ersoy, Hasan Furkan. Distinguished subspaces in topological sequence spaces theory[J]. AIMS Mathematics, 2020, 5(4): 2858-2868. doi: 10.3934/math.2020183

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Abstract

In this paper, we study $R_{\lambda}$-semiconservative $FK$-spaces for Riesz-method defined by the Riesz matrix $(R)$ and give some characterizations. We show that if $\ell_{A}$ is $\ell$-replaceable, then $A$ can not be $R_{\lambda}$-semiconservative and also if $X_{A}$ is $R_{\lambda}$-conull $FK$-space then it must be $R_{\lambda}$-semiconservative space. In addition, we determine a new $r(\lambda)$ and $rb(\lambda)$ type duality of a sequence space $X$ containing $\varphi$. The paper aims to develop some new subspaces which each one has its own value on topological sequence spaces theory. These subspaces are called as $R_{\lambda}S; R_{\lambda}W; R_{\lambda}F^{+}; $ and $R_{\lambda}B^{+}$ for a locally convex $FK$-space X containing $\varphi$. The subspaces mentioned in the work requires some serious studies and they arose independently from the literature which was done at the recent stage of the development of summability through functional analysis.

References

[1]	J. Boos, Classical and Modern Methods in Summability, Oxford University Press, New York, Oxford, 2000.
[2]	A. Wilansky, Summability Through Funtional Analysis, North Holland, 1984.
[3]	A. Wilansky, Functional Analysis, Blaisdell Press, 1964.
[4]	K. Zeller, Allgemeine Eigenschaften von Limitierungsverfahren, Math. Z., 53 (1951), 463-487. doi: 10.1007/BF01175646
[5]	K. Zeller, Theorie der Limitierungsverfahren, Berlin-Heidelberg New York, 1958.
[6]	I. Dağadur, C_λ-conull FK-spaces, Demonstr. Math., 35 (2002), 835-848.
[7]	J. A. Osikiewicz, Equivalance results for Cesàro submethods, Analysis, 20 (2000), 35-43. doi: 10.1524/anly.2000.20.1.35
[8]	S. A. Mohiuddine, A. Alotaibi, Weighted almost convergence and related infinite matrices, J. Inequal. Appl., 2018 (2018), 15.
[9]	A. K. Snyder, A. Wilansky, Inclusion Theorems and Semiconservative FK spaces, Rocky Mtn. J. Math., 2 (1972), 595-603. doi: 10.1216/RMJ-1972-2-4-595
[10]	H. G. Ince, Cesàro semiconservative FK-Spaces, Math. Communs., 14 (2009), 157-165.
[11]	I. Dağadur, C_λ semiconservative FK-Spaces, Ukr. Math. J., 64 (2012), 908-918. doi: 10.1007/s11253-012-0697-y
[12]	U. Değer, On Approximation by Nörlund And Riesz Submethods in Variable Expotent Lebesgue Spaces, Commun. Fac. Sci. Univ. Ank. Series A1, 67 (2018), 46-67.
[13]	M. Buntinas, Convergent and bounded Cesaro sections in FK-space, Math. Z., 121 (1971), 191-200. doi: 10.1007/BF01111591
[14]	G. Goes, S. Goes, Sequence of bounded variations and sequences of Fourier coefficients I., Math. Z., 118 (1970), 93-102. doi: 10.1007/BF01110177
[15]	M. Temizer Ersoy, B. Altay, H. Furkan, On Riesz Sections in Sequence Spaces, Adv. Math. Compt. Sci., 24 (2017), 1-10. doi: 10.9734/JAMCS/2017/35815
[16]	M. S. Macphail, C. Orhan, Some Properties of Absolute Summability Domains, Analysis, 9 (1989), 317-322. doi: 10.1524/anly.1989.9.4.317
[17]	E. Malkowsky, F. Başar, A Survey On Some Paranormed Sequence Spaces, Filomat, 31 (2017), 1099-1122. doi: 10.2298/FIL1704099M
[18]	A. Malkowsky, F. Özger, V. Veličković, Matrix Transformations on Mixed Paranorm Spaces, Filomat, 31 (2017), 2957-2966. doi: 10.2298/FIL1710957M
[19]	H. B. Ellidokuzoglu, S. Demiriz, Euler-Riesz Difference Sequence Spaces, Turk. J. Math. Comput. Sci., 7 (2017), 63-72.
[20]	F. Gökçe, M. A. Sarıgöl, Generalization of the Absolute Cesàro Space and Some Matrix Transformations, Numer. Func. Anal. Opt., 40 (2019), 1039-1052. doi: 10.1080/01630563.2019.1596130

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