Matrix transformations over non-Archimedian fields (Q1089537)

Let K be a non-trivial non-Archimedian field and let K be complete under the metric of valuation. Let \(x=\{x_ n\}\) be a sequence defined over K. Defining the sequence spaces m(K), \(\ell (K)\), \(\Gamma\) (K) and \(\Gamma^*(K)\) over the field K, the author considers the matrix transformations of the form \(y_ n=\sum^{\infty}_{p=1}a_{np}x_ p.\) By means of functional analysis techniques, the author obtains the necessary and sufficient conditions on the infinite matrix \(A=(a_{np})\) in order that the matrix should transform (i) \(\Gamma\) (K) into \(\Gamma^*(K)\); (ii) \(\ell (K)\) into \(\Gamma^*(K)\); (iii) \(\Gamma\) (K) into m(K); (iv) m(K) into \(\Gamma^*(K)\).