Domain of the Nörlund Matrix in Some of Maddox’s Spaces

Abstract

Maddox defined the sequence spaces $\ell _{\infty }(p)$, c(p) and $c_{0}(p)$ in Maddox (Q J Math Oxf Lond 18(2):345–355, 1967) and Maddox (Proc Comb Phil Soc 64:335–340, 1968), respectively. In the present paper, following Yeşilkayagil and Başar (Abstr Appl Anal, 2014), the Nörlund sequence spaces $\ell _{\infty }\left( N^{t},p\right)$, $c\left( N^{t},p\right)$ and $c_{0}\left( N^{t},p\right)$ of non-absolute type are introduced and it is proved that those sequence spaces are linearly isomorphic to the spaces $\ell _{\infty }(p)$, c(p) and $c_{0}(p)$, respectively. The $\alpha$-, $\beta$- and $\gamma$-duals of the spaces $\ell _{\infty }(N^{t},p)$, $c(N^{t},p)$ and $c_{0}(N^{t},p)$ are determined and the bases of the spaces $c(N^{t},p)$ and $c_{0}(N^{t},p)$ are given. Besides this, the classes of matrix transformations from $\ell _{\infty }(N^{t},p)$ to $\ell _{\infty }$, f, $f_{0}$, c, $c_{0}$ and from $\lambda (p)$ to $\mu (N^{t},p)$ are characterized, where $\lambda ,\mu$ denote any of the classical sequence spaces $\ell _{\infty }$, c or $c_{0}$.