Non-trivial zero decompositions in the theory of representative systems (Q1317574)

A sequence \(\{x_ k\}^ \infty_{k=1}\) in a complete separable l.c. space \(H\) is called an ARS (absolutely representative system) if any \(x\in H\) can be written as an absolutely convergent series: \(x= \sum^ \infty_{k=1} c_ k x_ k\). The sequence \(\{x_ k\}^ \infty_{k=1}\) is said to admit an NZD (non-trivial zero decomposition) if there is some absolutely convergent series \(\sum^ \infty_{n=1} a_ k x_ k= 0\), but not all \(a_ k= 0\). In this paper the author uses techniques previously developed in earlier papers (see references of current paper) to obtain more extensive results on the connection between the properties ARS and NZD. The theorems and their proofs are given in terms of complex function theory and are too complicated to be repeated here.