On sequence spaces defined by arithmetic function and Hausdorff measure of noncompactness

Abstract

We construct an infinite matrix $\mathfrak{𝔖}=(s_{nk})$ defined by

for all $n,k=1,2,3,\dots$, where $S(n)$ stands for the sum of the positive divisors of $n$, and introduce sequence spaces ${\ell }_p(\mathfrak{𝔖})$, $c_0(\mathfrak{𝔖})$, $c(\mathfrak{𝔖})$ and ${\ell }_{\infty }(\mathfrak{𝔖})$ by employing the matrix $\mathfrak{𝔖}$, where . We construct Schauder bases and compute $\alpha$-, $\beta$- and $\gamma$- duals of the newly constructed spaces. We state and prove characterization theorems related to matrix transformation from the spaces ${\ell }_p(\mathfrak{𝔖})$, $c_0(\mathfrak{𝔖})$, $c(\mathfrak{𝔖})$ and ${\ell }_{\infty }(\mathfrak{𝔖})$ to the spaces ${\ell }_{\infty }$, $c$, $c_0$ and ${\ell }_1$. Finally, we determine essential conditions for compactness of a matrix operator from the sequence space $X\in \left\{{\ell }_p(\mathfrak{𝔖}),c_0(\mathfrak{𝔖}),c(\mathfrak{𝔖}),{\ell }_{\infty }(\mathfrak{𝔖})\right\}$ to any of the sequence spaces ${\ell }_{\infty }$, $c$, $c_0$ or ${\ell }_1$.