On Some Difference Sequence Spaces of Weighted Means and Compact Operators‎

Abstract

‎In the peresent paper‎, ‎by using generalized weighted mean and difference‎ ‎matrix of order $m,$ we introduce the sequence spaces $X(u,v,\Delta^{(m)})$‎, ‎where $X$ is one of the spaces $\ell_{\infty}$‎, ‎$c$ or $c_{0}$‎. ‎Also‎, ‎we‎ ‎determine the $\alpha$-‎, ‎$\beta$‎- ‎and $\gamma$-duals of those spaces and‎ ‎construct their Schauder bases for $X\in\{c,c_{0}\}$‎. ‎Morever‎, ‎we give the‎ ‎characterization of the matrix mappings on the spaces $X(u,v,\Delta^{m})$‎ ‎ for $X\in\{\ell_{\infty},c,c_{0}\}.$ Finally‎, ‎we characterize some‎ ‎classes of compact operators on the spaces $\ell_{\infty}(u,v,\Delta^{m})$ and‎ ‎$c_{0}(u,v,\Delta^{m})$ by using the Hausdorff measure of noncompactness‎.