\(l^\infty(X)-l^p(Y)\) summability of mapping matrices (Q536465)

Let \(X,Y\) be Banach spaces, \(\lambda(X)\) (resp., \(\mu(Y)\)) be an \(X\)-valued (resp., \(Y\)-valued) vector space and \(T_{ij}:X\rightarrow Y\) linear operators for \(i,j\in\mathbb{N}\). The infinite matrix \((T_{ij})\) maps \(\lambda(X)\) into \(\mu(Y)\) if, for each \(i\) and \(\{x_{j}\}\in\lambda(X)\), the series \(\sum_{j=1}^{\infty}T_{ij}x_{j}\) converges and \(\{\sum_{j=1}^{\infty}T_{ij}x_{j}\}_{i}\in\mu(Y)\). One of the important problems is to characterize such matrix-maps between classical vector-valued sequence-spaces. In this paper, the authors give characterizations of matrix-maps from \(l^{\infty}(X)\) into \(l^{p}(Y)\), \(p\geq1\), and actually consider the case where the operators \(T_{ij}\) are not linear operators but belong to a class of operators which they call dissecting. The results are interesting even for the case of linear operators. The results are quite technical and will not be described here. The interested reader is referred to the paper.