Paranormed difference sequence spaces in the domain of Ces\`{a}ro means of order $\alpha$
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The purpose of this paper essentially deals with the sequence spaces $\ell_\infty(\mathcal{C}_\alpha, \Delta^n, q)$, $c(\mathcal{C}_\alpha, \Delta^n, q)$ and $c_0(\mathcal{C}_\alpha, \Delta^n, q)$ defined as the domain of composition of Ces\`{a}ro matrix $\mathcal{C}_\alpha$ of order $\alpha$ and the difference matrix $\Delta^n$ of order $n$ in the paranormed Maddox's sequence spaces $\ell_\infty(q), c(q)$ and $c_0(q)$, respectively. In addition, investigations have been performed to calculate their topological properties, inclusion relations, determine $\alpha-, \beta-$ and $\gamma-$ duals, and construct the Schauder basis of the new spaces. Finally, we emphasize the characterization of certain matrix mappings between the new described sequence spaces.
Abstract
The purpose of this paper essentially deals with the sequence spaces $\ell_\infty(\mathcal{C}_\alpha, \Delta^n, q)$, $c(\mathcal{C}_\alpha, \Delta^n, q)$ and $c_0(\mathcal{C}_\alpha, \Delta^n, q)$ defined as the domain of composition of Ces\`{a}ro matrix $\mathcal{C}_\alpha$ of order $\alpha$ and the difference matrix $\Delta^n$ of order $n$ in the paranormed Maddox's sequence spaces $\ell_\infty(q), c(q)$ and $c_0(q)$, respectively. In addition, investigations have been performed to calculate their topological properties, inclusion relations, determine $\alpha-, \beta-$ and $\gamma-$ duals, and construct the Schauder basis of the new spaces. Finally, we emphasize the characterization of certain matrix mappings between the new described sequence spaces.