Measures of Noncompactness in $\bar{N}(p,q)$ Summable Sequence Spaces

TL;DR

This paper introduces the $ar{N}(p,q)$ summable sequence spaces, explores their properties, and characterizes the compactness of linear operators on these spaces using Hausdorff measure of noncompactness.

Contribution

It defines new sequence spaces $ar{N}(p,q)$ and provides criteria for matrix mappings and compactness of operators on these spaces.

Findings

Characterization of matrix transformations on $ar{N}(p,q)$ spaces.

Necessary and sufficient conditions for operator compactness.

Application of Hausdorff measure of noncompactness to these spaces.

Abstract

In this paper we first define the $\bar{N}(p,q)$ summable sequence spaces and obtain some basic results related to these spaces. The necessary and sufficient conditions for infinite matrices $A$ to map these spaces on $X~~,~~X=c_0, c \text{ or } \ell_{\infty}$ is obtained and Hausdorff measure of noncompactness is then used to obtain the necessary and sufficient conditions for the compactness of linear operators defined on these spaces.