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.
📌 3 Key Takeaways
🎯
Objective & Hypothesis: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.
📊
Key Findings & Data: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.
💡
Conclusion & Impact: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.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Absolute summability · Compact operator · Hausdroff meausures of noncompactness
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.
GÖKÇE, F. (2021). Compact and Matrix Operators on the Space $\left\vert \overline N_p^{\phi }\right\vert _{k}$. Fundamental Journal of Mathematics and Applications, Cilt: 4 - Sayı: 2, 124–133. https://ajindex.com/articles/759296
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
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
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
F. Gökçe, M.A. Sarıgöl, Some matrix and compact operators of the absolute Fibonacci series spaces, Kragujevac J. Math., 44(2) (2020), 273–286
F. Gökçe, M.A. Sarıgöl, Series spaces derived from absolute Fibonacci summability and matrix transformations, Boll. Unione Mat. Ital., 13(1) (2020), 29-38
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
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
M.A. Sarıgöl, Matrix transformations on fields of absolute weighted mean summability, Studia Sci. Math. Hungar., 48(3) (2011), 331-341
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
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
M.A. Sarıgöl, On the local properties of factored Fourier series, Appl. Math. Comput., 216(11) (2010), 3386-3390
W.T. Sulaiman, On summability factors of infinite series, Proc. Amer. Math. Soc., 115(2) (1992), 313- 317
E. Malkowsky, V. Rakocevic, An introduction into the theory of sequence space and measures of noncompactness, Zb. Rad.(Beogr), 9(17) (2000), 143-234
M. Stieglitz, H. Tietz, Matrix transformationen von folgenraumen, Eine Ergebnis¨ubersicht. Math. Z. 154(1) (1977), 1-16
I.J.Maddox, Elements of functinal analysis, Cambridge University Press, London, New York, 1970
M.A. Sarıgöl, Extension of Mazhar’s theorem on summability factors, Kuwait J. Sci., 42(3) (2015), 28-35
I. Djolovic, E. Malkowsky, Matrix transformations and compact operators on some new mth-order difference sequences, Appl. Math. Comput., 198(2) (2008), 700-714
L.S. Goldenstein, I.T. Gohberg, A.S. Markus, Investigations of some properties of bounded linear operators with the