On some generalized Cesàro difference sequence spaces (Q2708401)

For a fixed \(m\in\mathbb{N}\), the author considers the Cesàro difference sequence spaces NEWLINE\[NEWLINEC_p(\Delta^m):= \Biggl\{(x_k)\mid\Biggl( \sum^\infty_{n=1} \Biggl|{1\over n} \sum^n_{k=1} \Delta^m x_k\Biggr|^p\Biggr)< \infty\Biggr\}\quad (1\leq p<\infty),NEWLINE\]NEWLINE NEWLINE\[NEWLINEC_\infty(\Delta^m):= \Biggl\{(x_k)\mid \sup_{n\in\mathbb{N}}\Biggl|{1\over n} \sum^n_{k=1} \Delta^m x_k\Biggr|< \infty\Biggr\},NEWLINE\]NEWLINE where \(\Delta^1 x_k:= x_k- x_{k+1}\) and \(\Delta^m x_k:= \Delta^{m- 1}x_k- \Delta^{m-1} x_{k+1}\). For \(m=2\), these spaces were studied by \textit{Mursaleen}, \textit{M. A. Khatib} and \textit{Quamaruddin} [Bull. Calculatta Math. Soc. 89, No. 4, 337-342 (1997; Zbl 0934.46006)]. It is shown that \(C_p(\Delta^m)\) and \(C_\infty(\Delta^m)\) are Banach spaces and the Köthe-Toeplitz \(\alpha\)-dual space of \(C_\infty(\Delta^m)\) is the sequence space \(\{(a_k)\mid \sum_k k^m|a_k|< \infty\}\). Finally, the author characterizes the matrix transformations \(A: E\to C_p(\Delta^m)\) \((1\leq p<\infty)\) and \(A: E\to C_\infty(\Delta^m)\), where \(E\) is one of the spaces \(\ell_\infty\) or \(c\).