Some more Steinhaus type theorems over valued fields. II (Q1014150)

The present paper is a continuation of [\textit{P.\,N.\thinspace Natarajan}, Ann.\ Math.\ Blaise Pascal 6, No.\,1, 47--54 (1999; Zbl 0943.47058)]. The author considers operators generated by some infinite matrices on sequence spaces, both over Archimedean and non-Archimedean fields. The results show the impossibility to obtain bounded operators (with certain additional conditions) simultaneously between different pairs of spaces. A typical result for the non-Archimedean case is as follows. Let \(c_0\) be the space of sequences tending to 0, let \(c\) be the space of convergent sequences, and let \(l_\infty\) be the space of bounded ones. If an infinite matrix defines a bounded operator \(A\) from \(c_0\) to \(c\), such that \(\lim (Ax)_n=\sum_{k=0}^\infty x_k\), \(x=\{ x_k\}\in c\), then the same infinite matrix does not generate a bounded operator from \(l_\infty\) to \(c\).