  For a semi-finite von Neumann algebra $\mathcal A$, we study the case of
equality in Young's inequality of s-numbers for a pair of $\tau$-measurable
operators $a,b$, and we prove that equality is only possible if $|a|^p=|b|^q$.
We also extend the result to unbounded operators affiliated with $\mathcal A$,
and relate this problem with other symmetric norm Young inequalities.
