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

Abstract

Abstract Recently, some new sequence spaces ℓ p ( A α ) $\ell _{p}(\mathfrak{A}^{\alpha })$ ( 0 < p < ∞ ) $(0< p<\infty )$ , c 0 ( A α ) $c_{0}(\mathfrak{A}^{\alpha })$ , c ( A α ) $c(\mathfrak{A}^{\alpha })$ , and ℓ ∞ ( A α ) $\ell _{\infty }(\mathfrak{A}^{\alpha })$ have been studied by Yaying et al. (Forum Math., 2024, https://doi.org/10.1515/forum-2023-0138 ) as matrix domains of A α = ( a n , v α ) $\mathfrak{A}^{\alpha }=(a_{n,v}^{\alpha })$ , where a m , v α = { v α ρ ( α ) ( m ) , v ∣ m , 0 , v ∤ m , $$ a_{\mathfrak{m},v}^{\alpha }=\left \{ \textstyle\begin{array}{c@{\quad}c@{\quad}c} \dfrac{v^{\alpha }}{\rho ^{(\alpha )}(\mathfrak{m})} & , & v\mid \mathfrak{m}, \\ 0 & , & v\nmid \mathfrak{m},\end{array}\displaystyle \right . $$ and ρ ( α ) ( m ) : = $\rho ^{(\alpha )}(\mathfrak{m}):=$ sum of the α th $\alpha ^{\text{th}}$ power of the positive divisors of m ∈ 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.

Keywords

• Arithmetic divisor sum function
• Sequence spaces
• Geometric properties