The author studies the difference operator

${\Delta}$ and its powers as operators on the space of sequences

${s}_{r}:=\{\left({x}_{n}\right)\in {\u2102}^{\infty}\mid {sup}_{n}\left\{\right|{x}_{n}|/{r}^{n}\}<\infty \}$ for some

$r>0$. Obviously these operators can be represented as infinite matrices and in case

$r=1$ the space

${s}_{1}$ is just the space of bounded sequences. In this paper, among other things, the spectra of such operators as maps from

${s}_{r}$ to

${s}_{r}$ are studied and characterizations of infinite matrix mappings from

${({\Delta}-\lambda I)}^{-1}{s}_{r}$ to

${s}_{r}$ are given.