Abstract. Let $m\in{\msbm N}$ and $u$ be a sequence of nonzero terms. If $x=(x_{k})_{k=0}^{\infty }$ is any sequence of complex numbers we write $\Delta ^{(m)}x$ for the sequence of the $m$--th order differences of $x$ and $\Delta ^{(m)}_{u}X=\{x=(x)_{k=0}^{\infty }\colon u\Delta ^{(m)}x\in X\} $ for any set $X$ of sequences. We determine the $\beta $--duals of the sets $\Delta ^{(m)}_{u}X$ for $X=c_{0},c,\ell_{\infty }$, and characterize some matrix transformations between these spaces $\Delta ^{(m)} X$. Furthermore, we deal with similar problems on matrix transformations, using a well chosen Banach algebra $S_{\tau }$, and consider perturbed matrices.

AMS Subject Classification (1991): 40H05, 46A45

Keyword(s): Sequence spaces, difference sequences, matrix transformations

Received June 26, 2004. (Registered under 5837/2009.)