Multiplicativity of \(l_ p\) norms for matrices. II (Q798740)

Let \(| A|_ p=(\sum| a_{ij}|^ p)^{1/p}\), 1\(\leq p\leq\infty \), the summation being taken over all 1\(\leq i\leq m\), 1\(\leq j\leq n\), be the \(l_ p\) norm of a complex \(m\times n\) matrix \(A=(a_{ij})\in C_{m\times n}\). The author proves results of the form: find the smallest possible constants \(\tau\) (m,k,n,p,q) and \(\sigma\) (m,k,n,p,q) such that \(| AB|_ p\leq\tau (m,k,n,p,q)| A|_ p| B|_ q, | AB|_ p\leq\sigma (m,k,n,p,q)| A|_ q| B|_ p\) for all \(A\in C_{m\times k}\), \(B\in C_{k\times n}\), which generalize earlier results by \textit{A. Ostrowski} [Math. Z. 63, 2-18 (1955, Zbl 0065.247)] and by the author and \textit{E. G. Straus} in part I of the paper [ibid. 52-53, 351-360 (1983; Zbl 0516.15025)].