Domain of the triple band matrix on some Maddox's spaces

Abstract

‎The sequence spaces $\ell_\infty(p)$‎, ‎$c(p)$ and $c_0(p)$ were‎ ‎introduced and studied by Maddox [Proc‎. ‎Cambridge Philos‎. ‎Soc‎. ‎64‎ ‎(1968)‎, ‎335-340]‎. ‎In the present paper‎, ‎we introduce the sequence‎ ‎spaces $\ell_\infty(B,p)$‎, ‎$c(B,p)$ and $c_0(B,p)$ of non-absolute‎ ‎type which are derived by the triple band matrix $B(r,s,t)$ and is‎ ‎proved that the spaces $\ell_\infty(B,p)$‎, ‎$c(B,p)$ and $c_0(B,p)$‎ ‎are paranorm isomorphic to the spaces $\ell_\infty(p)$‎, ‎$c(p)$ and‎ ‎$c_0(p)$; respectively‎. ‎Besides this‎, ‎the $\alpha$-‎, ‎$\beta$‎- ‎and‎ ‎$\gamma$-duals of the spaces $\ell_\infty(B,p)$‎, ‎$c(B,p)$ and‎ ‎$c_0(B,p)$ are computed and the bases of the spaces $c(B,p)$ and‎ ‎$c_0(B,p)$ are constructed‎. ‎Finally‎, ‎the matrix mappings from the‎ ‎sequence spaces $\lambda(B,p)$ to a given sequence space $\mu$ and‎ ‎from the sequence space $\mu$ to the sequence space $\lambda(B,p)$‎ ‎are characterized‎, ‎where $\lambda\in\{\ell_\infty,c,c_0\}$‎.