Let $|| = ||\_{p,q}$ be an operator norm on $\mathbb R^{n \times m}$. In General, $\|AB\|\le \|A\|\|B\|$. Is there some criterion on $A, B$ (at least for some operator norms) so that $\|AB\| = \lVert A \rVert \lVert B\rVert$?



When is the matrix norm multiplicative
At least, there is this important case: in $C^\*$-algebras, there is an involution and the norm has the property that
$$\|x\|=\|x^\*\|=\|xx^\*\|^{1/2}.$$
In the special case of ${\bf M}\_n({\mathbb C})$, this is
$$\|AA^\*\|=\|A\|\cdot\|A^\*\|$$
where the norm is the usual operator norm $\|\cdot\|\_{2,2}$.


Actually this formula extends to the case of rectangular matrices.


