On some geometric properties of sequence spaces of generalized arithmetic divisor sum function

Recently, some new sequence spaces ${\ell }_p({\mathfrak{A}}^{\alpha })$ $\ell _{p}(\mathfrak{A}^{\alpha })$ $(0<p<\mathrm{\infty })$ $(0< p<\infty )$ , $c_0({\mathfrak{A}}^{\alpha })$ $c_{0}(\mathfrak{A}^{\alpha })$ , $c({\mathfrak{A}}^{\alpha })$ $c(\mathfrak{A}^{\alpha })$ , and ${\ell }_{\mathrm{\infty }}({\mathfrak{A}}^{\alpha })$ $\ell _{\infty }(\mathfrak{A}^{\alpha })$ have been studied by Yaying et al. (Forum Math., 2024, https://doi.org/10.1515/forum-2023-0138) as matrix domains of ${\mathfrak{A}}^{\alpha }=(a_{n,v}^{\alpha })$ $\mathfrak{A}^{\alpha }=(a_{n,v}^{\alpha })$ , where $a_{\mathfrak{m},v}^{\alpha }=\{\begin{array}{ccc}\frac{v^{\alpha }}{{\rho }^{(\alpha )}(\mathfrak{m})}& ,& v\mid \mathfrak{m},\\ 0& ,& v\nmid \mathfrak{m},\end{array}$ \( a_{\mathfrak{m},v}^{\alpha }=\left \{ \textstyle\begin{array}{c@{\quad}c@{\quad}c} \dfrac{v^{\alpha }}{\rho ^{(\alpha )}(\mathfrak{m})} & , & v\mid \mathfrak{m}, \\ 0 & , & v\nmid \mathfrak{m},\end{array}\displaystyle \right . \) and ${\rho }^{(\alpha )}(\mathfrak{m}):=$ $\rho ^{(\alpha )}(\mathfrak{m}):=$ sum of the ${\alpha }^{\text{th}}$ $\alpha ^{\text{th}}$ power of the positive divisors of $\mathfrak{m}\in \mathbb{N}$ $\mathfrak{m}\in \mathbb{N}$ . They obtained their duals, matrix transformations and associated compact matrix operators for these matrix classes.

This article deals with some geometric properties of these sequence spaces.