Empty-cored sequences in Banach spaces (Q2711474)

Let \(X\) be a Banach space. For a sequence \(x=(\xi_n)\) with \(\xi_n\in X\), the Knopp core \(K(x)\) is the intersection of the closed convex hulls of the sets \(\{\xi_n, \xi_{n+1}, \dots\}\) \((n\in\mathbb{N})\). The authors show that if (a) \(x\) is (equivalent to) the unit vector basis of \(\ell^1\) or (b) \((\xi_n)\) is weakly Cauchy with no weak limit, then \(K(x)= \emptyset\) and \(K(Ax)= \emptyset\) for each regular matrix \(A=(a_{nk})\) with scalar entries. It is shown that \(X\) is reflexive if and only if every sequence on the unit sphere has nonempty Knopp core. The authors prove also that \(K(Ax) \subset K(x)\) if one of the following statements is satisfied: (i) \(A\) is regular and \((\xi_n)\) is weakly Cauchy, (ii) \(A\) is regular and positive and \(x\) is any sequence with \(\xi_n\in X\), (iii) \(A\) is regular with \(\lim_n \sum_k|a_{nk} |=1\) and \((\xi_n)\) is bounded.