Domain of the Cesàro mean of order $\alpha$ in Maddox's space $\ell(p)$

Medine Yeşilkayagil Savaşcı, Feyzi Başar

The sequence space $\ell(p)$ was defined by "I. J. Maddox, \emph{Spaces of strongly summable sequences}, Quart. J. Math. Oxford (2), extbf{18} (1967), 345--355". In the present paper, we introduce the paranormed Cesàro sequence space $\ell(C_{\alpha},p)$ of order $\alpha$, of non-absolute type as the domain of Cesàro mean $C_{\alpha}$ of order $\alpha$ and prove that the spaces $\ell(C_{\alpha},p)$ and $\ell(p)$ are linearly paranorm isomorphic. Besides this, we compute the $\alpha$-, $\beta$- and $\gamma$- duals of the space $\ell(C_{\alpha},p)$ and construct the basis of the space $\ell(C_{\alpha},p)$ together with the characterization of the classes of matrix transformations from the space $\ell(C_{\alpha},p)$ into the spaces $\ell_{\infty}$ of bounded sequences and $f$ of almost convergent sequences, and any given sequence space $Y$, and from a given sequence space $Y$ into the sequence space $\ell(C_{\alpha},p)$. Finally, we emphasize on some geometric properties of the space $\ell(C_{\alpha},p)$.