Equivalence constants for certain matrix norms. (Q1414162)

The best equivalence constants for \(| A| _{pq}\) and \(\| A\| _{pq}\) matrix norms are obtained. Here \(| A| _{pq}=(\sum^n_{j=1}(\sum^m_{i=1} | a_{ij}| ^p)^{q/p})^{1/q}\) and \(\| A\| _{pq}=\max \{| Ax| _q: | x| _p\leqslant 1\}\). Define \({\lambda _{pq}(n)=1}\) for \(p\geqslant q\) and \(\lambda _{pq}(n)= n^{1/p-1/q}\) for \(p<q\). The author shows that in the following main inequalities the constants are the best possible \[ | A| _{pq} \leqslant \lambda_{pr}(m) \lambda_{qs}(n) | A| _{rs}, \] \[ \| A\| _{pq} \leqslant \lambda_{qs}(m) \lambda_{rp} (n) \| A\| _{rs}, \] \[ \| A\| _{pq} \leqslant \lambda_{qr}(m) \lambda_{s'p}(n) | A| _{rs}, \] here \(1\leqslant p,q,r,s\leqslant \infty\), \(1/s+1/s'=1\). These results generalize inequalities of \textit{M. Goldberg} [Linear Multilinear Algebra 21, 173--179 (1987; Zbl 0636.15010)]. The question of determining the best constant~\(d\) in the inequality \(| A| _{pq}\leqslant d \| A\| _{rs}\) will be considered in the author's forthcoming paper.