4
$\begingroup$

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 $\tau$.

$\endgroup$

1 Answer 1

8
$\begingroup$

The necessary and sufficient condition is that $|x|^p$ and $|y^*|^q$ are proportional. This can be deduced from Dixmier's paper (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.

$\endgroup$
2
  • 3
    $\begingroup$ You probably mean that $|x|^p$ and $|y^*|^p$ are proportional. $\endgroup$ Jun 2, 2011 at 19:52
  • 1
    $\begingroup$ Right. Thanks, Mikael. I'll correct it right away. $\endgroup$ Jun 2, 2011 at 20:53

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.