ON SOME PROPERTIES OF DIFFERENCE OPERATOR WITH SOME CHARACTERIZATIONS

Abstract

In this paper, we introduce the generalized spaces of the form $\mathcal{H}(f,g,\Delta_n^{m})$, where $\mathcal{H}$ represents as one  of the spaces $\ell_{\infty}$, $c$ or $c_{0}$. The k\"{o}the duals corresponds to these spaces will be computed and construction of the  Schauder bases for $\mathcal{H}\in\{c,c_{0}\}$ will be given. Also, some matrix characterizations concerning these spaces will be computed. Moreover, the characterization of some classes of compact operators on the spaces $\ell_{\infty}(f,g,\Delta_n^{m})$ and $c_{0}(f,g,\Delta_n^{m})$ by employing the Hausdorff measure of non-compactness will be determined.