Some paranormed sequence spaces derived from generalized divisor sum function

Abstract

We explore the properties of the paranormed sequence spaces $c_0(p,\mathscr{D}^\alpha)$, $c(p,\mathscr{D}^\alpha)$, and $\ell_\infty(p,\mathscr{D}^\alpha)$, which are generated by an infinite matrix $\mathscr{D}^\alpha$ involving a generalized divisor sum function $\sigma^{(\alpha)}$ to classical Maddox spaces $c_0(p)$, $c(p)$, and $\ell_\infty(p)$, respectively. The matrix $\mathscr{D}^\alpha=(d^\alpha_{m,r})$ is defined such that $d^\alpha_{m,r} = \dfrac{r^\alpha}{\sigma^{(\alpha)}(m)}$ if $r$ is a divisor of $m$, and $0$, otherwise. Our analysis includes the determination of the Schauder basis and the computation of dual spaces ($\alpha$-, $\beta$-, and $\gamma$-duals) for these newly defined paranormed spaces. Additionally, we characterize matrix transformations from $\ell_{\infty}(p,\mathscr{D}^\alpha)$ into several known sequence spaces, and present related matrix characterizations as direct consequences.