Infinite matrices and almost convergent sequences (Q1908948)

Let \((p_k)\) be a sequence of real numbers with \(p_k> 0\). A sequence, whose \(k\)th term is \(x_k\), will be denoted by \(x\) or \((x_k)\). Let \[ \ell(p, s)= \Biggl\{x: \sum_k k^{-s} |x_k|^{p_k}< \infty,\;s\geq 0\Biggr\}. \] Let \(f\) be the space of all almost convergent sequences. Let \(f_\infty\) be the space of all \(f\)-bounded sequences. In this paper, the class \((\ell(p, s), f)\) of all infinite matrices transforming \(\ell(p, s)\) into \(f\), and the class \((\ell(p, s), f_\infty)\) of all infinite matrices transforming \(\ell(p, s)\) into \(f_\infty\) are characterized.