Banach-Steinhaus type theorems for statistical and \({\mathcal I}\)-convergence with applications to matrix maps
Let \({\mathcal I}\) be an ideal of subsets of \(\mathbb N\). By \textit{A. R. Freedman} and \textit{J. J. Sember} [Pac. J. Math. 95, 293--305 (1981; Zbl 0504.40002)], the ideal \({\mathcal I}\) is said to have the property (APO) if it contains all finite subsets of \(\mathbb N\) and for every sequence \((K_i)\) in \({\mathcal I}\) there exists a sequence \((L_i)\) in \({\mathcal I}\) such that the symmetric difference \(K_{i}\Delta L_{i}\) is finite for each \(i\) and \(\bigcup_{i}L_{i}\in{\mathcal I}\). A sequence \((x_k) \) in a normed space \(X\) is said to be \({\mathcal I}\)-convergent to \(l\in X\) if \(\{k\in\mathbb N\mid\|x_k-l\|\geq\varepsilon\} \in{\mathcal I}\) for each \(\varepsilon>0\). For a nonnegative regular summability matrix \(T=(t_{nk})\), the \(T\)-statistical convergence defined by the author in [Tartu Ül. Toimetised No 928, 41--52 (1991)] coincides with the \({\mathcal I}_T\)-convergence, where \({\mathcal I}_T= \{K\subset\mathbb N\mid\lim_n\sum_{k\in K}t_{nk}=0\} \) is an ideal with the property (APO). The main theorem of this paper is the following. Let \({\mathcal I}\) be an ideal with the property (APO) and let \((A_n) \) be a sequence of bounded linear operators from a separabel Banach space \(X\) to a Banach space \(Y\). Then \((A_nx)\) is boundedly \({\mathcal I}\)-convergent for each \(x\in X\) if and only if \(\sup_n \|A_{n}\|<\infty\) and \((A_n\varphi)\) is \({\mathcal I}\)-convergent for all \(\varphi\in \Phi\), where \(\Phi\) is a countable fundamental set of \(X\). From this theorem the author obtains a number of results concerning matrix transformations in vector valued sequence spaces.
scientific article; zbMATH DE number 5704114