  In the given paper we first introduce $\bar{N}_{\Delta^{-}}^{q}$ summable
difference sequence spaces and prove some properties of these spaces. We then
obtain the necessary and sufficient conditions for infinite matrices $A$ to map
these sequence spaces on the spaces $c_0, c$ and $\ell_\infty$, the Hausdorff
measure of noncompactness is then used to obtain the necessary and sufficient
conditions for the compactness of the linear operators defined on these spaces.
