A Note On The Matrix Operators Of Absolute Nörlund Series Space

Yıl 2023, Cilt: 11 Sayı: 1, 8 - 14, 30.04.2023

Fadime Gökçe

Öz

On the recent study, the series space $\left\vert N_{p}^{\theta }\right\vert (q )$ has been introduced as the set of all series summable by the absolute Nörlund summability method, which includes the Maddox's space $l(q )$, the absolute Cesaro space $\left\vert C_{\alpha }\right\vert _{k}$ and the absolute Nörlund space $\left\vertN_{p}^{\theta }\right\vert _{k}$, and studied in terms of some topological and algebraic properties and matrix transformations by Gökçe and Sarıgöl. In this paper, some characterizations of matrix operators between the absolute Nörlund series space $\left\vert N_{p}^{\theta }\right\vert (q )$ and the classical sequence spaces $c,c_0,l_{\infty}$ are given. Also, it is shown that the matrix operators are bounded operators. Finally, certain results are obtained as a special case.

Anahtar Kelimeler

Nörlund means;, matrix transformation;, bounded linear operator;, Maddox’s space.

Kaynakça

• [1] P. Z. Alp, M. I˙lkhan, On the difference sequence space$l_p (\hat T^q)$, Math. Sci. Appl. E-Notes Vol: 7, No. 2 (2019), 161-173.
• [2] T. M. Flett, On an extension of absolute summability and some theorems of Littlewood and Paley, Proc. London Math. Soc. Vol: 7 (1957), 113-141.
• [3] F. G¨okc¸e, Compact and Matrix Operators on the Space $\left\vert\overline N_p^{\phi}\right\vert _ {k} $. FUJMA. Vol: 4 No. 2 (2021), 124-133.
• [4] F. G¨okc¸e, M.A. Sarıg¨ol, Some matrix and compact operators of the absolute Fibonacci series spaces. Kragujevac J. Math. Vol: 44, No.2, (2020), 273–286.
• [5] F. G¨okc¸e, M.A. Sarıg¨ol, Series spaces derived from absolute Fibonacci summability and matrix transformations. Boll. Unione Mat. Ital. Vol: 13, (2020), 29-38.
• [6] F. G¨okc¸e, M.A. Sarıg¨ol, On absolute Euler spaces and related matrix operators, Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci. Vol: 90 No:5 (2020), 769-775.
• [7] F. G¨okc¸e, M.A. Sarıg¨ol, Generalization of the absolute Ces`aro space and some matrix transformations, Num. Func. Anal. Opt. Vol: 40, (2019), 1039-1052.
• [8] F. G¨okc¸e, M.A. Sarıg¨ol, Extension of Maddox’s space l(m) with N¨orlund means, Asian-Eur. J. Math. Vol: 12 No.6 (2019), 1-12.
• [9] F. G¨okc¸e, M.A. Sarıg¨ol, A new series space$\left\vert \overline{N}_{p}^{\theta }\right\vert \left(\mu \right) $ and matrix transformations with applications, Kuwait J. Sci. Vol:45 No:4 (2018), 1-8.
• [10] F. G¨okc¸e, M.A. Sarıg¨ol, Generalization of the space l(p) derived by absolute Euler summability and matrix operators. J. Ineq. Appl. Vol: 2018 (2018), 133
• [11] K.G. Grosse-Erdmann, Matrix transformations between the sequence spaces of Maddox, J. Mathe.Anal, Appl., Vol: (1993), 223–238.
• [12] G. C. Hazar, M. A. Sarıg¨ol, On Absolute N¨orlund Spaces and Matrix Operators, Acta Math. Sinica, English Series Vol: 34 (2018), 812-826.
• [13] M. Ilkhan, Matrix Domain of a Regular Matrix Derived by Euler Totient Function in the Spaces c0 and c, Mediterr. J. Math.,Vol: 17, No.1 (2020), 1-21.
• [14] I. J. Maddox, Some properties of paranormed sequence spaces, J. London Math. Soc. Vol: 1, (1969), 316-322.
• [15] I. J. Maddox, Paranormed sequence spaces generated by infinite matrices, Math. Proc. Cambridge Philos. Soc. Vol: 64, (1968), 335-340.
• [16] I. J. Maddox, Spaces of strongly summable sequences, The Quart. J. Math. Vol: 18, (1967), 345-355.
• [17] E. Malkowsky, V. Rakocevic, On matrix domains of triangles. Appl.Math. Comp. Vol:189, No. 2, (2007), 1146-1163.
• [18] E. Malkowsky, V. Rakocevic, An introduction into the theory of sequence space and measures of noncompactness, Zb. Rad.(Beogr), Vol:9, No.17, (2000), 143-234.
• [19] M.A. Sarıg¨ol, Spaces of series summable by absolute Ces`aro and matrix operators, Comm. Math. Appl. Vol:7 (2016), 11-22.
• [20] A. Wilansky, Summability Through Functional Analysis, Math. Studies, 85. North Holland , Amsterdam, (1984).