Matrix transformations of \(X_ p\) and \(Z_ p\) into \(V_ \sigma\) (Q1184832)

Let \(V_ \sigma\) denote the space of bounded sequences all of whose invariant means are equal. \textit{E. Savas} [Math. Stud. 59, No. 1-4, 170- 176 (1991)] obtained a necessary and sufficient condition for an infinite matrix which transforms \(X_ p\) into the space of almost convergent sequences, where \[ X_ p=\left\{x\in X:\;\left(\sum_{n=1}^ \infty \left| {1\over n}\sum_{i=1}^ n x_ i\right|^ p\right)^{1/p}<\infty\right\} \] with the norm \(\| x\|_ p=(\sum_{n=1}^ \infty |{1\over n} \sum_{i=1}^ n x_ i|^ p)^{1/p}\) and \(X\) is the set of all real sequences \(x=(x_ k)\). In this paper the author generalizes this result by obtaining a characterization of matrices in the space \((X_ p,V_ \sigma)\). He further establishes a similar result for the matrices in the space \((Z_ p,V_ \sigma)\) where \(Z_ p=\{x\in X\): \(\sup_ n |{1\over n}\sum_{i=1}^ n x_ i|^ p\}\).