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

Abstract

In this 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 into the spaces $c,~c_0,$ and $\ell_{\infty}$. Finally, 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.