Unconditional Cauchy series and uniform convergence on matrices (Q1766405)

Motivated by general versions of the Hahn--Schur summation theorem and the well-known matrix theorem of Antosik and Swartz, the authors investigate uniform convergence on matrices with coefficients in a given normed space. The investigations are based on a new characterization of unconditional Cauchy series in terms of certain separation properties of subfamilies of \({\mathcal P}({\mathbb N})\) given in Theorem 2.1. Let \(\sum_i x_i\) be a series in \(a\) normed space \(X\,.\) The following properties are equivalent: (i) \(\sum_i x_i\) is an unconditional Cauchy series. (ii) For any pair \([(A_i)_i,(B_i)_i]\) of disjoint sequences of mutually disjoint finite subsets of \({\mathbb N},\) there exist \(B\subseteq {\mathbb N}\) and an infinite set \(M\subseteq{\mathbb N}\) such that \(\sum_{i\in B} x_i\) is Cauchy, \(A_i\subseteq B\) and \(B_i\subseteq {\mathbb N}\setminus B\) for \(i\in M.\) There are two main results on uniform convergence on matrices (with coefficients in a given normed space): the first one (Theorem 3.1) is a version of the Hahn--Schur summation theorem due to \textit{Ch.~Swartz} [Southeast Asian Bull.\ Math.\ 20, No.~2, 57--58 (1996; Zbl 0858.40006)] that improves Theorem 1 by \textit{J.--D.\ Wu} and \textit{S.--J.\ Lu} [J.\ Math.\ Anal.\ Appl.\ 257, No.~1, 29--38 (2001; Zbl 0988.40003)]; the second one (Theorem 3.2) contains the mentioned matrix theorem of Antosik and Swartz as an easy corollary.