Some Euler Spaces of Difference Sequences and Matrix Mappings

Abstract

Altay and Ba\c{s}ar [2] and Altay, Ba\c{s}ar and Mursaleen [3] introduced the Euler sequence spaces $e^r_{0\ ,}{\ e}^r_{c}$, and $e^r_{\infty \ ,}$ respectively. Polat and Ba\c{s}ar [25] introduced the spaces $e^r_0$(${\Delta}^{(m)}$), $e^r_{c\ \ }$(${\Delta }^{(m)}$), and $e^r_{\infty \ }$(${\Delta }^{(m)}$) consisting of all sequences whose $m^{th}$ order differences are in the Euler spaces $e^r_{0,}\ e^r_{c}$, and $e^r_{\infty \ ,}$ respectively. Moreover, the authors give some topological properties and inclusion relations, and determine the $\alpha$-, $\beta$-, $\gamma$-, and continuous duals of the spaces $e^r_0\ ,$ $e^r_{c\ \ },e^r_p,$ $e^r_{\infty \ },$ $e^r_0$(${\Delta }^{(m)}$), $e^r_{c\ \ }$(${\Delta }^{(m)}$) and $e^r_{\infty \ }$(${\Delta }^{(m)}$), for $1\le p<\infty $ and their basis have been constructed. The last section of the article is devoted to the characterization of some matrix mappings on the sequence spaces $e^r_{c\ \ }\ $and $e^r_{c\ \ }\mathrm{(}{\Delta }^{(m)}\mathrm{)\ }$.