Tensorial transformations of some matrix spaces (Q1261249)

Let \(\Omega\) be the vector space of all infinite matrices over the field of scalars \(K\). The author considers some subspaces of \(\Omega\) denoted by \(\partial\), \(\ell_{pp}\) and defined by \[ \partial=\Bigl\{a: a\in \Omega,\;| a_{mn}|^{{1\over m+n}}\longrightarrow 0,\text{ as }m+n\longrightarrow\infty\Bigr\}. \] \[ \ell_{pp}=\left\{a: a\in\Omega,\;\sum\sum| a_{mn}|^ p=\lim \sum_{N\to\infty}\sum_{0\leq m+n<N} | a_{mn}|^ p,\;1\leq p<\infty\right\}. \] The author proves some tensorial transformation of such matrix spaces, as the following. Theorem 1. If \(g= \partial^{rs}_{mn}\) is a tensor of order four from \(\ell_{pp}\) to \(\partial\) defined by \[ a= g.b= \sum_{r+s\geq 0} \sum \partial^{rs}_{mn} b_{rs}= a_{mn}, \] where \(b=(b_{rs})\in\ell_{pp}\), then \(a=(a_{mn})\in \partial\) if and only if \[ \left(\sum\sum_{r+s\geq 0}|\partial^{rs}_{mn}|^ q\right)^{{1\over m+n}}\longrightarrow 0\quad\text{as }m+n\longrightarrow\infty \] uniformly in \(r\), \(s\) where \(p>1\), \({1\over p}+{1\over q}= 1\). The article contains some more similar theorems concerning some other matrix subspaces of \(\Omega\).