Some Paranormed Difference Sequence Spaces of Order $m$ Derived by Generalized Means and Compact Operators

TL;DR

This paper introduces a new paranormed sequence space derived from generalized means and difference operators, analyzes its properties, duals, and matrix transformations, and investigates operator norms and measures of noncompactness.

Contribution

It defines a novel sequence space using generalized means and difference operators, proves its completeness and basis, and characterizes its duals and matrix transformations.

Findings

The space is complete under a suitable paranorm.

The space has a Schauder basis.

Necessary and sufficient conditions for matrix transformations are established.

Abstract

We have introduced a new sequence space $l(r, s, t, p ;\Delta^{(m)})$ combining by using generalized means and difference operator of order $m$. We have shown that the space $l(r, s, t, p ;\Delta^{(m)})$ is complete under some suitable paranorm and it has Schauder basis. Furthermore, the $\alpha$-, $\beta$-, $\gamma$- duals of this space is computed and also obtained necessary and sufficient conditions for some matrix transformations from $l(r, s, t, p; \Delta^{(m)})$ to $l_{\infty}, l_1$. Finally, we obtained some identities or estimates for the operator norms and the Hausdorff measure of noncompactness of some matrix operators on the BK space $l_{p}(r, s, t ;\Delta^{(m)})$ by applying the Hausdorff measure of noncompactness.