Summability characterization for B-convex and super reflexive spaces (Q762723)

In this paper the following was proved: Theorem: A Banach space X is B- convex (resp. super reflexive) if and only if for every double sequence \(\{x^ m_ n\}\) in the unit ball of X, there is a lower triangular (resp. lower triangular and sign ordered) absolutely convex matrix \(T=(C_{mn})\) such that \[ (I)\quad \limsup\quad \| t_ m\| <1\quad where\quad t_ m=\sum^{m}_{n=1}C_{mn}x^ m_ n. \] It was observed that (I) can be replaced by \[ (II)\quad \lim \inf \quad \| t_ m\| <1. \] In the B-convex case, it was further observed that (II) can be replaced by \[ (III)\quad \lim \inf \quad \| t_ m\| =0. \] The paper concludes with an open problem: Can we replace (II) by (III) in the super reflexive case? Subsequently, this problem is answered in the affirmative independently by \textit{S. V. R. Naidu} [Dept. of Applied Mathematics, A.U.P.G. Extension Centre, Nuzvid, 521 201, India] in a private talk and \textit{J. R. Partington} [Fitzwilliam College, Cambridge, CB30DG, England] in a private communication.