On The Difference Sequence Space $l_p(\hat{T}^q)$

Yıl 2019, Cilt: 7 Sayı: 2, 161 - 173, 15.10.2019

Merve İlkhan , Pınar Zengin Alp

https://doi.org/10.36753/mathenot.597703

Cited By: 15

Öz

In this study, we introduce a new matrix $\hat{T}^q=(\hat{t}^q_{nk})$ by

\[

\hat{t}^q_{nk}=\left \{

\begin{array}

[c]{ccl}%

\frac{q_n}{Q_n} t_n & , & k=n\\

\frac{q_k}{Q_n}t_k-\frac{q_{k+1}}{Q_n} \frac{1}{t_{k+1}} & , & k<n\\

0 & , & k>n .

\end{array}

\right.

\]

where $t_k>0$ for all $n\in\mathbb{N}$ and $(t_n)\in c\backslash c_0$. By using the matrix $\hat{T}^q$, we introduce the sequence space $\ell_p(\hat{T}^q)$ for $1\leq p\leq\infty$. In addition, we give some theorems on inclusion  relations associated with $\ell_p(\hat{T}^q)$ and  find the $\alpha$-, $\beta$-, $\gamma$- duals of this space. Lastly, we analyze the necessary and sufficient conditions for an infinite matrix to be in the classes $(\ell_p(\hat{T}^q),\lambda)$ or $(\lambda,\ell_p(\hat{T}^q))$, where $\lambda\in\{\ell_1,c_0,c,\ell_\infty\}$.

Anahtar Kelimeler

sequence spaces, matrix transformations, Schauder basis, $\alpha- \beta- \gamma-duals

Kaynakça

• [1] Bektaş, Ç.A. Et, M. and Çolak, R., Generalized difference sequence spaces and their dual spaces, J. Math. Anal.Appl., 292(2), 423-432, 2004.
• [2] Aydın, C. and Başar, F., Some new sequence spaces which include the spaces `p and `1, Demonstratio Math.,38(3), 641-656, 2005.
• [3] Mursaleen, M., Başar, F. and Altay, B., On the Euler sequence spaces which include the spaces `p and `1 II,Nonlinear Anal. TMA, 65(3), 707–717, 2006.
• [4] Kirişçi, M. and Başar, F., Some new sequence spaces derived by the domain of generalized difference matrix,Comput. Math. Appl., 60(A2), 1299-1309, 2010.
• [5] Polat, H., Karakaya, V. and Şimşek, N., Difference sequence spaces derived by using a generalized weightedmean, Appl. Math. Lett., 24(5), 608-614, 2011.
• [6] Mursaleen, M. and Noman, A. K., On some new sequence spaces of non-absolute type related to the spaces $\ell_p$ and $\ell_\infty$ I, Filomat, 25(2),33-51, 2011.
• [7] Savaş, E., Matrix transformations between some new sequence spaces, Tamkang J. Math., 19(4), 75-80, 1988.
• [8] Candan, M., A new sequence space isomorphic to the space l(p) and compact operators, J. Math. Comput. Sci.,4(2), 306-334 2014.
• [9] Kara, E. E. and İlkhan, M., Some properties of generalized Fibonacci sequence spaces, Linear MultilinearAlgebra, 64(11), 2208-2223, 2016.
• [10] Demiriz, S. and Duyar, O., Domain of the generalized double Cesaro matrix in some paranormed spaces ofdouble sequences, Tbil. Math. J. 10(2), 43-56, 2017.
• [11] Altay, B. and Kama, R., On Cesaro summability of vector valued multiplier spaces and operator valued series,Positivity 22(2), 575-586, 2018.
• [12] Das, A. and Hazarika, B., Some new Fibonacci difference spaces of non-absolute type and compact operators,Linear Multilinear Algebra, 65(12), 2551-2573 2017.
• [13] Hazar, G.C. and Sarıgöl, M. A., On absolute Nörlund spaces and matrix operators, Acta Math. Sin. (Engl. Ser.)34(5), 812-826, 2018.
• [14] Hazar, G.C. and Sarıgöl, M. A., Absolute Cesàro series spaces and matrix operators, Acta Appl. Math., 154(1),153-165, 2018.
• [15] Gökçe, F. and Sarıgöl, M. A., Generalization of the absolute Cesàro space and some matrix transformations,Numer. Funct. Anal. Optim., 40(9), 1039–1052, 2019.
• [16] İlkhan, M. and Kara, E. E. A new Banach space defined by Euler Totient matrix operator, Oper. Matrices, 13(2),527-544, (2019).
• [17] Başar, F., Summability Theory and Its Applications, Bentham Science Publishers, İstanbul, 2012.
• [18] Kızmaz, H., On certain sequence spaces, Can. Math. Bull., 24(2), 169-176, 1981.
• [19] Et, M. and Çolak, R., On some generalized difference sequence spaces, Soochow J. Math., 21, 377-386, 1995.
• [20] Altay, B. and Başar, F., The matrix domain and the fine spectrum of the difference operator $\Delta$ on the sequencespace $\ell_p$, $0 < p < 1$, Commun. Math. Anal., 2(2), 1-11, 2007.
• [21] Başar, F. and Altay, B., On the space of sequences of p-bounded variation and related matrix mappings, Ukr.Math. J., 55, 136–147, 2003.
• [22] Çolak, R. and Et, M. and Malkowsky, E., Some topics of Sequence Spaces, Lecture Notes in Mathematics, FıratUniv. Press, Fırat Univ. Elazığ, Turkey, 1-63, 2004.
• [23] Kara, E. E., Some topological and geometrical properties of new Banach sequence spaces, J. Inequal. Appl., 38,2013.
• [24] Altay, B., Başar, F., On the paranormed Riesz sequence spaces of non-absolute type. Southeast Asian Bulletin ofMathematics, 26(5), 701–715, 2003.
• [25] Kara, E. E. and İlkhan, M., On some Banach sequence spaces derived by a new band matrix, Br. J. Math. Comput.Sci., 9(2), 141-159, 2015.
• [26] Sarıgöl, M. A., On difference sequence spaces, J. Karadeniz Tech. Univ., Fac. Arts Sci., Ser. Math.-Phys, 10, 63-71,1987.
• [27] Et, M., On some difference sequence spaces, Turk. J. Math., 17, 18-24, 1993.
• [28] Sönmez, A, Almost convergence and triple band matrix, Math. Comput. Model., 57, 2393-2402, 2013.
• [29] Mursaleen, M. and Noman, A. K., On some new difference sequence spaces of non-absolute type, Math.Comput. Model., 52, 603-617, 2010.
• [30] Altay, B. and Ba¸sar, F. and Mursaleen, M., On the Euler sequence spaces which include the spaces $\ell_p$ and $\ell_\infty$ I,Inform. Sci., 176(10), 1450-1462, 2006.
• [31] Aydın, C. and Başar, F., Some new difference sequence spaces, Appl. Math. Comput., 157(3),677-693, 2004.
• [32] Candan, M, Domain of the double sequential band matrix in the classical sequence spaces, J. Inequal. Appl.,281, 2012.
• [33] Candan, M, Almost convergence and double sequential band matrix, Acta Math. Sci., 34B(2), 354-366, 2014.
• [34] Stieglitz, M. and Tietz, H., Matrix transformationen von folgenraumen eine ergebnisübersicht, 154, 1-16, 1977.
• [35] Maddox, I. J., Lecture Notes in Mathematics. Infinite Matrices of Operators, Springer-Verlag, Berlin HeidelbergNew York, 1980. (ISBN-3-540-09764-3)
• [36] Altay, B. and Başar, F., Certain topological properties and duals of the matrix domain of a triangle matrix in asequence space, J. Math. Anal. Appl., 336(1), 632-645, 2007.