This is mostly a rehashing James Lee’s excellent lecture notes on the proof of Lieb’s concavity theorem, which discusses things beyond the proof.
In order to prove this, we require this other theorem:
For those who’ve forgotten, we say that a map is operator concave if given any we have
At a first glance, it seems to be quite strange that Theorem would imply Theorem .
To prove this proposition, we require the following lemma,
This is not that difficult to prove. See Lee’s notes for an elementary proof. I also think that this could be proven using elementary arguments from Löewner theory, maybe another post about it? Now, armed with this lemma, let us Proposition
Proof of Lemma . Note that from the Stieltjes representation, ◻