New sequence spaces derived by using generalized arithmetic divisor sum function and compact operators

Abstract

Abstract Define an infinite matrix D α = ( d n , v α ) \mathfrak{D}^{\alpha}=(d^{\alpha}_{n,v}) by d n , v α = { v α σ ( α ) ⁢ ( n ) , v ∣ n , 0 , v ∤ n , d^{\alpha}_{n,v}=\begin{cases}\dfrac{v^{\alpha}}{\sigma^{(\alpha)}(n)},&v\mid n,\\ 0,&v\nmid n,\end{cases} where σ ( α ) ⁢ ( n ) \sigma^{(\alpha)}(n) is defined to be the sum of the 𝛼-th power of the positive divisors of n ∈ N n\in\mathbb{N} , and construct the matrix domains ℓ p ⁢ ( D α ) \ell_{p}(\mathfrak{D}^{\alpha}) ( 0 < p < ∞ 0<p<\infty ), c 0 ⁢ ( D α ) c_{0}(\mathfrak{D}^{\alpha}) , c ⁢ ( D α ) c(\mathfrak{D}^{\alpha}) and ℓ ∞ ⁢ ( D α ) \ell_{\infty}(\mathfrak{D}^{\alpha}) defined by the matrix D α \mathfrak{D}^{\alpha} . We develop Schauder bases and determine 𝛼-, 𝛽- and 𝛾-duals of these new spaces. We characterize some matrix transformation from ℓ p ⁢ ( D α ) \ell_{p}(\mathfrak{D}^{\alpha}) , c 0 ⁢ ( D α ) c_{0}(\mathfrak{D}^{\alpha}) , c ⁢ ( D α ) c(\mathfrak{D}^{\alpha}) and ℓ ∞ ⁢ ( D α ) \ell_{\infty}(\mathfrak{D}^{\alpha}) to ℓ ∞ \ell_{\infty} , 𝑐, c 0 c_{0} and ℓ 1 \ell_{1} . Furthermore, we determine some criteria for compactness of an operator (or matrix) from X ∈ { ℓ p ⁢ ( D α ) , c 0 ⁢ ( D α ) , c ⁢ ( D α ) , ℓ ∞ ⁢ ( D α ) } X\in\{\ell_{p}(\mathfrak{D}^{\alpha}),c_{0}(\mathfrak{D}^{\alpha}),c(\mathfrak{D}^{\alpha}),\ell_{\infty}(\mathfrak{D}^{\alpha})\} to ℓ ∞ \ell_{\infty} , 𝑐, c 0 c_{0} or ℓ 1 \ell_{1} .