Measures of Noncompactness in $\left(\bar{N}_{Δ^{-}}^{q}\right)$ Summable Difference Sequence Spaces

Abstract: 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.