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Properties of some sets of sequences and application to the spaces of bounded difference sequences of order $\mu$. (English) Zbl 1016.40002
The author studies the difference operator ${\Delta }$ and its powers as operators on the space of sequences ${s}_{r}:=\left\{\left({x}_{n}\right)\in {ℂ}^{\infty }\mid {sup}_{n}\left\{|{x}_{n}|/{r}^{n}\right\}<\infty \right\}$ 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 ${\left({\Delta }-\lambda I\right)}^{-1}{s}_{r}$ to ${s}_{r}$ are given.
##### MSC:
 40C05 Matrix methods in summability 46A45 Sequence spaces 47A10 Spectrum and resolvent of linear operators
##### Keywords:
difference operator; space of sequences; spectra