The Norm of Certain Matrix Operators On the New Block Sequence Space

Abstract

The purpose of the this study is to introduce the sequence space $$ \ell_{p}(E,B(r,s))=\bigg\{x=(x_{n})\in \omega: \sum_{n=1}^{\infty} \bigg|\sum_{j\in E_n}rx_{j}+\sum_{j\in E_{n+1}}sx_{j}\bigg|^{p}<\infty\bigg\}, $$ where $E=(E_n)$ is a partition of finite subsets of the positive integers, $r,s\in \mathbb{R}\backslash \{0\}$ and $p\geq 1$. The topological and algebraical properties of this space are examined. Furthermore, we establish some inclusion relations. Finally, the problem of finding the norm of certain matrix operators such as Copson and Hilbert from $\ell_p$ into $\ell_{p}(E,B(r,s)) $ is investigated.

Keywords

• Capson operators,Hilbert operators,Matrix domain

References

1. [1] M. Kirişçi, F. Başar, Some new sequence spaces derived by the domain of generalized difference matrix, Comput. Math. Appl., 60 (2010), 1299-1309.
2. [2] B. Altay, F. Başar, The fine spectrum and the matrix domain of the difference operator $\Delta$ on the sequence space $\ell_p , (0
3. [3] F. Başar, B. Altay, On the space of sequences of p-bounded variation and related matrix mappings, Ukr. Math. J., 55(1) (2003), 136-147.
4. [4] F. Başar, B. Altay, M. Mursaleen, Some generalizations of the space bvp of p- bounded variation sequences,Nonlinear Anal., 68(2) (2008), 273-287.
5. [5] M. Candan, Domain of the double sequential band matrix in the classical sequence spaces, J. Inequal. Appl., 1 (2012), 281.
6. [6] M. Candan, E. E. Kara, A study on topological and geometrical characteristics of new Banach sequence spaces, Gulf J. Math., 3(4)(2015), 67-84.
7. [7] M. Candan, Domain of the double sequential band matrix in the spaces of convergent and nul sequences, Adv. Difference Equ., 1 (2014), 163.
8. [8] M. Candan, A new sequence space isomorphic to the space `(p) and compact operators, Journal of Mathematical and Computational Science, 4(2) (2014), 306-334.

9. [9] F. Başar, Summability Theory and Its Applications, Bentham Science Publishers, Istanbul, Turkey, 2012.
10. [10] D. Foroutannia, On the block sequence space `p(E) and related matrix transformations, Turk. J. Math., 39 (2015), 830-841.
11. [11] H. Roopaei, D. Foroutannia, The norm of certain matrix operators on the new difference sequence spaces I, Sahand Communications in Mathematical Analysis (SCMA), 3(1)(2016), 1-12.
12. [12] G. H. Hardy, J. E. Littlewood, G. Polya, Inequalities, 2nd edition, Cambridge University Press, Cambridge, 2001.
13. [13] D. Foroutannia, Upper bound and lower bound for matrix operators on weighted sequence spaces, Ph.D. Thesis, Zahedan, 2007.
14. [14] S. Erfanmanesh, D. Foroutannia, Some new semi-normed sequence spaces of non-absolute type and matrix transformations, TWMS J. Pure Appl. Math., 4(2) (2015), 96-108.
15. [15] S. Erfanmanesh, D. Foroutannia, Generalizations of Kothe-Toeplitz duals and null duals of new difference sequence spaces, J. Contemp. Math. Anal., 51(3) (2016), 125-133.