"Some $m$th-order Difference Sequence Spaces of Generalized Means and Compact Operators"

TL;DR

This paper introduces new sequence spaces based on generalized means and difference operators of order m, analyzes their structure, duals, and characterizes compact operators acting on them.

Contribution

It defines and studies the properties of new sequence spaces involving generalized means and difference operators, including their bases, duals, and compact operators.

Findings

Spaces are complete normed linear spaces.

Spaces c0 and c have Schauder bases.

Necessary and sufficient conditions for matrix transformations and compact operators are established.

Abstract

In this paper, new sequence spaces $X(r, s, t ;\Delta^{(m)})$ for $X\in \{l_\infty, c, c_0\}$ defined by using generalized means and difference operator of order $m$ are introduced. It is shown that these spaces are complete normed linear spaces and the spaces $c_0(r, s, t ;\Delta^{(m)})$, $c(r, s, t ;\Delta^{(m)})$ have Schauder basis. Furthermore, the $\alpha$-, $\beta$-, $\gamma$- duals of these spaces are computed and also obtained necessary and sufficient conditions for some matrix transformations from $X(r, s, t ;\Delta^{(m)})$ to $X$. Finally, some classes of compact operators on the spaces $c_0(r, s, t ;\Delta^{(m)})$ and $l_{\infty}(r, s, t ;\Delta^{(m)})$ are characterized by using the Hausdorff measure of noncompactness.