"Some $m$th-order Difference Sequence Spaces of Generalized Means and Compact Operators"

Abstract

In this paper, new sequence spaces $X(r,s,t;{\Delta }^{(m)})$ for $X\in \{l_{\infty },c,c_0\}$ defined by using generalized means and difference operator of order $m$ are introduced. It is shown that these spaces are complete normed linear spaces and the spaces $c_0(r,s,t;{\Delta }^{(m)})$, $c(r,s,t;{\Delta }^{(m)})$ have Schauder basis. Furthermore, the $\alpha$-, $\beta$-, $\gamma$- duals of these spaces are computed and also obtained necessary and sufficient conditions for some matrix transformations from $X(r,s,t;{\Delta }^{(m)})$ to $X$. Finally, some classes of compact operators on the spaces $c_0(r,s,t;{\Delta }^{(m)})$ and $l_{\infty }(r,s,t;{\Delta }^{(m)})$ are characterized by using the Hausdorff measure of noncompactness.

arXiv comment18 pages. arXiv admin note: substantial text overlap with arXiv:1307.5817, arXiv:1307.6208, arXiv:1307.5884