The Generalized Difference Operator Δ i 3 of Order Three and Its Domain in the Sequence Spaces ℓ 1 and b v

Abstract

Most recently, the generalized difference operator ${\mathrm{\Delta }}_i^3$ of order three was defined and its domain in Hahn sequence space $h$ was calculated. In this paper, the spaces ${\ell }_1\left({\mathrm{\Delta }}_i^3\right)$ and $bv\left({\mathrm{\Delta }}_i^3\right)$ are introduced as the domain of generalized difference operator ${\mathrm{\Delta }}_i^3$ of order three in the sequence spaces ${\ell }_1$ and $bv$ . Then, some topological properties of ${\ell }_1\left({\mathrm{\Delta }}_i^3\right)$ and $bv\left({\mathrm{\Delta }}_i^3\right)$ are given, and some inclusion relations are shown. Additionally, algebraic dual, $\alpha -$ , $\beta -$ , and $\gamma -$ dual spaces of ${\ell }_1\left({\mathrm{\Delta }}_i^3\right)$ and $bv\left({\mathrm{\Delta }}_i^3\right)$ are computed. In the last section, the classes $\left(\mu \left({\mathrm{\Delta }}_i^3\right):\lambda \right)$ and $\left(\lambda :\mu \left({\mathrm{\Delta }}_i^3\right)\right)$ of matrix transformations are characterized, where $\mu =\left\{{\ell }_1,bv\right\}$ and $\lambda =\left\{c,c_0,{\ell }_1,{\ell }_{\infty },bs,cs,bv,h\right\}$ .