Matrix summability methods and weakly unconditionally Cauchy series
Let \(X\) be a normed space and \(A=[a_{ij}]\) be an infinite matrix with real values. A sequence \(\{x_j\}\) in \(X\) has \(A\) limit \(x\) if \(\lim_i\sum _{j=1}^\infty a_{ij}x_j=x\) and we write \(A-\lim x_j=x\). A series \(\sum_jx_j\) in \(X\) is weakly unconditionally Cauchy (wuc) if \(\sum _{j=1}^\infty |x'(x_j)|<\infty\) for every \(x'\in X'\). Let \(\sum x_j\) be a (formal) series in \(X\); the authors consider the space \(S_A(\sum_jx_j)= \{\{t_j\}\in l^\infty: A-\lim\sum_{j=1}^n t_jx_j\) exists\} (with the sup-norm) and study the relationship between wuc series and the space \(S_A (\sum_jx_j)\). They show that if \(X\) is complete, \(\sup_i\sum_{j=1}^\infty |a_{ij}|=M>0\) and \(\sum_jx_j\) is wuc, then \(S_A (\sum_j x_j)\) is complete. Moreover, if \(A\) is regular and \(X\) is complete, then \(\sum_j x_j\) is wuc iff \(S_A (\sum_jx_j)\) is complete. They also consider the linear operator \(T:S_A (\sum_j x_j)\to X\) defined by \(T\{t_{j}\}=A-\lim\sum_{j=1}^{n} t_jx_j\) and show that \(T\) is continuous iff \(\sum_{j}x_{j}\) is wuc. Similar results are given for the weak analogue of the space \(S_A (\sum_j x_j)\) defined by \(S_{wA}(\sum_j x_j)= \{t_j\}\in l^\infty: \text{weak}-A-\lim \sum_{j=1}^n t_j x_j\) exists\}. Finally, the authors establish a version of the Orlicz-Pettis Theorem for series which have a weak unconditional \(A\) limit and \(A\) is a regular matrix.
scientific article; zbMATH DE number 5566175