Compactness of binomial difference operator of fractional order and sequence spaces

Abstract

In this article we introduce binomial difference sequence spaces of fractional order $\alpha ,$$b_0^{r,s}\left( \Delta ^{(\alpha )}\right) ,$$b_c^{r,s}\left( \Delta ^{(\alpha )}\right)$ and $b_{\infty }^{r,s}\left( \Delta ^{(\alpha )}\right)$ by employing fractional difference operator $\Delta ^{(\alpha )},$ defined by $\Delta ^{(\alpha )}x_k=\sum \limits _{i=0}^{\infty }(-1)^i\frac{\Gamma (\alpha +1)}{i!\Gamma (\alpha -i+1)}x_{k-i}.$ We give some topological properties, obtain the Schauder basis and determine the $\alpha -,$$\beta -$ and $\gamma -$ duals of the spaces. We characterize the matrix classes $(b_c^{r,s}(\Delta ^{(\alpha )}),\ell _p),$$(b_c^{r,s}(\Delta ^{(\alpha )}),\ell _{\infty })$ and $(b_c^{r,s}(\Delta ^{(\alpha )}),c).$ We characterize certain classes of compact operators on the space $b_c^{r,s}(\Delta ^{(\alpha )})$ using Hausdorff measure of non-compactness. Finally, we present the graphical interpretation of the operator $B^{r,s}\left( \Delta ^{(\alpha )}\right)$.