Let $1\leq p,q,r\leq \infty$ such that $\frac{1}{r}=\frac{1}{p}+\frac{1}{q}$. Let $S\_p$ denote the Schatten space. For any $x\in S\_p$ and any $y\in S\_q$ we have
$$
||xy||\_{S\_r} \leq ||x||\_{S\_p}||y||\_{S\_q}
$$
(noncommutative Hölder's inequality).


Does it exists necessary and sufficient conditions on $x,y$ in order to have an equality in this inequality?


More generally, I ask the same question replacing $S\_p$ by the noncommutative $L\_p$-space $L\_p(M)$ associated with a semifinite von Neumann algebra $M$ equipped with a normal semifinite faithful trace $\tau$.



equality in noncommutative Hölder inequality
The necessary and sufficient condition is that $|x|^p$ and $|y^\*|^q$ are proportional. This can be deduced from [Dixmier's paper](http://archive.numdam.org/article/BSMF_1953__81__9_0.pdf) (although it is not clearly stated that way there; it is based on Proposition 8). It probably also appears in more modern treatments (Nelson, Terp, Haagerup, Hiai, Kosaki, etc.) but I don't have the sources here to check that. 


