On a theorem of Robinson for non-archimedian Banach spaces (Q1911877)

The object of this paper is to obtain the non-Archimedean operator analogue of Kojima-Schur theorem for Banach spaces due to \textit{A. Robinson} [Proc. Lond. Math. Soc., II. Ser. 52, 132-160 (1950; Zbl 0039.06203)]. Let \(K\) be a field complete with respect to a non-Archimedean valuation. Let \(X\) be a non-Archimedean Banach space and \(B(X)\) denote the space of bounded linear operators on \(X\) into \(X\). Let \(A= (A_{np})\), where \(n, p= 1,2,3,\dots\) be an infinite matrix such that \(A_{np}\in B(X)\) for all \(n\) and \(p\) and \(I\) denote the identity operator on \(X\). Let \((T_n)\) be a sequence in \(B(X)\) with \(\|(T_n)\|= \sup_{n\geq 1} \| T_nx_n\|\) for each \(x_n\in X\) with \(\| x_n\|\leq 1\) and \(y_n= \sum^\infty_{p= 1}A_{np} x_p\), \(n= 1,2,3,\dots\) be transformations. Then \(A= (A_{np})\) is said to be convergence preserving if \((y_n)\) converges whenever \((x_n)\) converges in the norm. \(A\) is said to be regular if \(x_p\to s\) as \(p\to \infty\) implies \(y_n\to s\) as \(n\to\infty\). Robinson's theorem for an infinite matrix of bounded linear operators on a non-Archimedean Banach space \(X\) into itself proved in the paper is the following: Theorem: For the matrix \(A= (A_{np})\) to be regular, it is necessary and sufficient that for each \(x\in X\), \(\lim_{n\to\infty} A_{np}x= 0\) for each fixed \(p\), \(\lim_{n\to\infty} \sum^\infty_{p=1} A_{np}x= Ix\), \(\lim_{p\to\infty} A_{np}x= 0\) for each fixed \(n\), and \(\sup_{\substack{ 1\leq p<\infty\\ 1\leq n<\infty}} \| A_{np}\|\leq M\), where \(M\) is a constant independent of \(n\) and \(p\).