Some Paranormed Difference Sequence Spaces of Order $m$ Derived by Generalized Means and Compact Operators

Abstract

We have introduced a new sequence space $l(r, s, t, p ;Δ^{(m)})$ combining by using generalized means and difference operator of order $m$. We have shown that the space $l(r, s, t, p ;Δ^{(m)})$ is complete under some suitable paranorm and it has Schauder basis. Furthermore, the $α$-, $β$-, $γ$- duals of this space is computed and also obtained necessary and sufficient conditions for some matrix transformations from $l(r, s, t, p; Δ^{(m)})$ to $l_{\infty}, l_1$. Finally, we obtained some identities or estimates for the operator norms and the Hausdorff measure of noncompactness of some matrix operators on the BK space $l_{p}(r, s, t ;Δ^{(m)})$ by applying the Hausdorff measure of noncompactness.

Please withdraw this paper as there are some logical gap in some results. 20 pages. arXiv admin note: substantial text overlap with arXiv:1307.5883, arXiv:1307.5817, arXiv:1307.5884