Matrix transformations involving generalized almost convergent sequences (Q2330050)

Let \(p=\left(p_{k}\right) \) be a positive real sequence. In addition to the usual Maddox sequence spaces \[\ell \left(p\right) =\left\{ \left(x_{k}\right) :\sum_{k}\left\vert x_{k}\right\vert ^{p_{k}}<\infty \right\} ,\] \[c_{0}\left(p\right) =\left\{\left(x_{k}\right) :\lim_{k}\left\vert x_{k}\right\vert ^{p_{k}}=0\right\},\;\; c\left(p\right) =c_{0}\left(p\right) \oplus \left\langle \left(1,1,\dots\right) \right\rangle \] and \[\ell_{\infty }\left(p\right) =\left\{\left(x_{k}\right) :\sup_{k}\left\vert x_{k}\right\vert ^{p_{k}}<\infty\right\} ,\] the authors consider the spaces \[M_{0}\left(p\right) =\bigcup\limits_{N\in \mathbb{N}}\left\{ \left(x_{k}\right) :\sum_{k}\left\vert x_{k}\right\vert N^{-1/p_{k}}<\infty \right\} ,\] \[f_{0}\left(p\right) =\left\{ \left(x_{k}\right) :\lim_{m\rightarrow \infty }\left\vert \frac{1}{m+1}\sum_{i=0}^{m}x_{n+i}\right\vert =0\; {\text{uniformly}}\; {\text{in}}\; n\right\} \] and \[f\left(p\right) =f_{0}\left(p\right) \oplus \left\langle \left(1,1,\dots \right) \right\rangle ,\] which are perhaps not so familiar. They characterize the matrix transformations \(A:E\rightarrow F\), where \(E\in \left\{ c_{0}\left(p\right) ,c\left(p\right) ,\ell ^{\infty }\left(p\right) ,M_{0}\left(p\right) ,\ell \left(p\right) ,bv\right\} \) and \(F\in \left\{ f_{0}\left(p\right) ,f\left(p\right) \right\}\). In so doing, they correct some errors in two papers of \textit{S. Nanda}, see [Mat. Vesn., N. Ser. 13(28), 305--312 (1976; Zbl 0358.46010)] and [J. Indian Math. Soc., New Ser. 40, 173--184 (1976; Zbl 0445.40003)].