Document Type: Research Paper

**Authors**

Department of Mathematics, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran.

**Abstract**

In the present paper, we introduce the sequence space \[{l_p}(E,\Delta) = \left\{ x = (x_n)_{n = 1}^\infty : \sum_{n = 1}^\infty \left| \sum_{j \in {E_n}} x_j - \sum_{j \in E_{n + 1}} x_j\right| ^p < \infty \right\},\] where $E=(E_n)$ is a partition of finite subsets of the positive integers and $p\ge 1$. We investigate its topological properties and inclusion relations. Moreover, we consider the problem of finding the norm of certain matrix operators from $l_p$ into $ l_p(E,\Delta)$, and apply our results to Copson and Hilbert matrices.

**Keywords**

**Main Subjects**

[1] B. Altay and F. Basar, *The ne spectrum and the matrix domain of the dierence operator Δ on the sequence space lp, (0 < p < 1)*, Commun. Math. Anal., 2(2) (2007) 1-11.

[2] F. Basar, *Summability Theory and Its Applications, Bentham Science Publishers*, e-books, Monographs, Istanbul, 2012.

[3] F. Basar and B. Altay, *On the space of sequences of p-bounded variation and related matrix mappings*, Ukr. Math. J., 55(1) (2003) 136-147.

[4] F. Basar, B. Altay, and M. Mursaleen, *Some generalizations of the space bvp of p-bounded variation sequences*, Nonlinear Anal., 68(2) (2008) 273-287.

[5] D. Foroutannia, *On the block sequence space lp(E) and related matrix transfor- mations*, Turk. J. Math., 39 (2015) 830-841.

[6] D. Foroutannia, *Upper bound and lower bound for matrix opwrators on weighted sequence spaces*, Doctoral dissertation, Zahedan, 2007.

[7] G.H. Hardy, J.E. Littlewood, and G. Polya, *Inequalities*, 2nd edition, Cambridge University press, Cambridge, 2001.

[8] G.J.O. Jameson and R. Lashkaripour, *Norms of certain operators on weighted lp spaces and Lorentz sequence spaces*, J. Inequal. Pure Appl. Math., 3(1) (2002) Article 6.

[9] H. Kizmaz, *On certain sequence spaces I*, Canad. Math. Bull., 25(2) (1981) 169-176.

[10] R. Lashkaripour and J. Fathi, *Norms of matrix operators on bvp*, J. Math. Inequal., 6(4) (2012) 589-592.

[11] M. Mursaleen and A.K. Noman, *On some new dierence sequence spaces of non-absolute type*, Math. Comput. Modelling, 52 (2010) 603-617.

[12] H. Roopaei and D. Foroutannia, *The norm of certain matrix operators on the new dierence sequence spaces*, preprint.