Some $B$-Difference Sequence Spaces Derived by Using Generalized Means and Compact Operators

TL;DR

This paper introduces new generalized difference sequence spaces, studies their properties, duals, and matrix transformations, and characterizes compact operators on these spaces using measures of noncompactness.

Contribution

It defines new sequence spaces using generalized means and difference operators, analyzes their structure, duals, and compact operators, extending existing sequence space theory.

Findings

Spaces are complete paranormed spaces.

Spaces for c(p), c_0(p), l(p) have Schauder basis.

Characterization of compact operators via Hausdorff measure.

Abstract

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