Sequences of operators in Köthe-Toeplitz duals (Q1970266)

Let \(X\) and \(Y\) denote non-Archimedean Banach spaces over a complete nontrivially non-Archimedean valued field \(K\) and \(B(X,Y)\) the Banach algebra of bounded linear operators from \(X\) to \(Y\) with the operator norm. The following result is established among others: Let \(S\) be the set of all sequences \(\{T_k\}\) such that the group norm is finite. Then \(S\) is a non-Archimedean Banach space with the natural operations under the norm \(\|\{T_k\}\|\). Let \(\{A_k\}\) be a sequence of operators from \(X\) to \(Y\). Characterizations for \(\{A_k\}\) to belong to \(\alpha\) and \(\beta\) duals of \(X\)-valued sequence spaces \(c_0(X)\), \(c(X)\), \(\ell_\infty(X)\) and \(\ell_p(X)\) with \(1\leq p<\infty\) are also given.