Compact and Matrix Operators on the Space $\left\vert \overline N_p^{\phi }\right\vert _{k}$

Year 2021, Volume: 4 Issue: 2, 124 - 133, 01.06.2021

Fadime Gökçe

https://doi.org/10.33401/fujma.882309

Cited By: 4

Abstract

In this paper, determining the operator norm, we give certain characterizations of matrix transformations from the space $ \left\vert \overline N_p^{\phi }\right\vert _{k}$, the space of all series summable by the absolute weighted mean summability method, to one of the classical sequence spaces $c_{0},c,l_{\infty }.$ Also, we obtain the necessary and sufficient conditions for each matrix in these classes to be compact and establish a number of estimates or identities for the Hausdorff measures of noncompactness of the matrix operators in these classes.

Keywords

Absolute summability , Compact operator , Hausdroff meausures of noncompactness , Matrix transformations , Operator norm , Sequence spaces , Weighted mean

References

• [1] M. Ilkhan, Matrix domain of a regular matrix derived by Euler Totient function in the spaces c0 and c, Mediterr. J. Math., 17(1) (2020), 1-21.
• [2] E. E. Kara, M. Bas¸arır, On compact operators and some Euler B(m)-difference sequence spaces, J. Math. Anal. Appl., 379(2) (2011), 499-511.
• [3] T.M. FLett, On an extension of absolute summability and some theorems of Littlewood and Paley, Proc. Lond. Math. Soc., 3(1) (1957), 113-141.
• [4] F. Gökçe, M.A. Sarıgöl, Some matrix and compact operators of the absolute Fibonacci series spaces, Kragujevac J. Math., 44(2) (2020), 273–286.
• [5] F. Gökçe, M.A. Sarıgöl, Series spaces derived from absolute Fibonacci summability and matrix transformations, Boll. Unione Mat. Ital., 13(1) (2020), 29-38.
• [6] F. Gökçe, M.A. Sarıgöl, On absolute Euler spaces and related matrix operators, Proc. Nat. Acad. Sci. India Sect. A, 90(5) (2020), 769-775.
• [7] F. Gökçe, M.A. Sarıgöl, Generalization of the absolute Ces`aro space and some matrix transformations, Numer. Funct. Anal. Optim., 40(9) (2019), 1039-1052.
• [8] M.A. Sarıgöl, Matrix transformations on fields of absolute weighted mean summability, Studia Sci. Math. Hungar., 48(3) (2011), 331-341.
• [9] M.A. Sarıgöl, Necessary and sufficient conditions for the equivalence of the summability methods jN; pnjk and jC;1jk, Indian J. Pure Appl. Math., 22(6) (1991), 483-489.
• [10] R. N. Mohapatra, M.A. Sarıgöl, On matrix operators on the series space j¯Nq p jk, Ukrainian Math. J., 69(11) (2018), 1772-1783.
• [11] M.A. Sarıgöl, On the local properties of factored Fourier series, Appl. Math. Comput., 216(11) (2010), 3386-3390.
• [12] W.T. Sulaiman, On summability factors of infinite series, Proc. Amer. Math. Soc., 115(2) (1992), 313- 317.
• [13] E. Malkowsky, V. Rakocevic, An introduction into the theory of sequence space and measures of noncompactness, Zb. Rad.(Beogr), 9(17) (2000), 143-234.
• [14] M. Stieglitz, H. Tietz, Matrix transformationen von folgenraumen, Eine Ergebnis¨ubersicht. Math. Z. 154(1) (1977), 1-16.
• [15] I.J.Maddox, Elements of functinal analysis, Cambridge University Press, London, New York, 1970.
• [16] M.A. Sarıgöl, Extension of Mazhar’s theorem on summability factors, Kuwait J. Sci., 42(3) (2015), 28-35.
• [17] I. Djolovic, E. Malkowsky, Matrix transformations and compact operators on some new mth-order difference sequences, Appl. Math. Comput., 198(2) (2008), 700-714.
• [18] L.S. Goldenstein, I.T. Gohberg, A.S. Markus, Investigations of some properties of bounded linear operators with their q-norms, Uchen. Zap. Kishinev. Gos. Univ., 29 (1957), 29-36.
• [19] M. Mursaleen, A.K. Noman, Compactness of matrix operators on some new difference sequence spaces, Linear Algebra Appl., 436(1) (2012), 41-52.
• [20] M. Mursaleen, A.K. Noman, Compactness by the Hausdorff measure of noncompactness, Nonlinear Anal., 73(8) (2010), 2541-2557.
• [21] E. Malkowsky, V. Rakocevic, Measure of noncompactness of linear operators between spaces of sequences that are (¯N ;q) summable or bounded, Czechoslovak Math. J., 51(3) (2001), 505-522.
• [22] G. C. Hazar Güleç, Applications of measure of noncompactness in the series spaces of generalized absolute Ces`aro means, KFBD, 10(1),(2020) 60-73.
• [23] G. C. Hazar Güleç, Compact matrix operators on absolute Ces`aro spaces, Numer. Funct. Anal. Optim., 41(1) (2020), 1-15.
• [24] E. Malkowsky, Compact matrix operators between some BK􀀀 spaces, in: M. Mursaleen (Ed.), Modern Methods of Analysis and Its Applications, Anamaya Publ., New Delhi, (2010), 86-120.
• [25] M.A. Sarıgöl, Norms and compactness of operators on absolute weighted mean summable series, Kuwait J. Sci., 43(4) (2016), 68-74.
• [26] V. Rakocevic, Measures of noncompactness and some applications, Filomat, 12(2) (1998), 87-120.
• [27] A.M. Jarrah, E. Malkowsky, Ordinary absolute and strong summability and matrix transformations, Filomat 17 (2003), 59-78.
• [28] E. Malkowsky, V. Rakocevic, On matrix domains of triangles, Appl. Math. Comput., 189(2) (2007), 1146-1163.