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

TL;DR

This paper introduces new summable difference sequence spaces, explores their properties, and characterizes the compactness of operators acting on these spaces using measures of noncompactness.

Contribution

It defines the $ar{N}_{ riangle^{-}}^{q}$ summable difference sequence spaces and provides criteria for operator compactness via measures of noncompactness.

Findings

Characterization of the sequence spaces $ar{N}_{ riangle^{-}}^{q}$.

Necessary and sufficient conditions for matrix mappings to classical sequence spaces.

Criteria for compactness of operators using Hausdorff measure of noncompactness.

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.