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I've previously asked this question on stack exchange.

I'm looking for a proof of the inequality

$$ \left[ \frac12(\left\|A+B\right\|_p^p + \left\|A-B\right\|_p^p)\right]^{2/p} \leq \left\|A\right\|_p^2 +C_p \left\|B\right\|_p^2 $$

where A and B are complex matrices of the same size, $\left\| .\right\|_p$ refers to the Schatten p-norm, p$\geq 2$, and $C_p=p-1$.

I have a reference for the opposite inequality when $1\leq p \leq 2$ but no mention of the one above. Are the two somehow equivalent? Am I missing an easy argument to follow one out of the other?

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  • $\begingroup$ I do mean $\geq$ instead of $\leq$. Sorry if it is worded strangely. Here is the reference just for good measure: link.springer.com/article/10.1007/BF01231769 The statement is that of Theorem 1. $\endgroup$
    – Florian Ente
    Commented Jan 7, 2021 at 15:45
  • $\begingroup$ Thanks for clarifying. No, the formulation is clear enough, just the fact itself confused me somehow. $\endgroup$
    – Christian Remling
    Commented Jan 7, 2021 at 16:01

1 Answer 1

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According to the Pisier-Xu survey "Non-commutative $L^p$ spaces" https://www.zbmath.org/?q=an%3A1046.46048, this is proved in

Ball, Keith; Carlen, Eric A.; Lieb, Elliott H. Sharp uniform convexity and smoothness inequalities for trace norms. Invent. Math. 115, No. 3, 463-482 (1994).

https://www.zbmath.org/?q=an%3A0803.47037

Edit: I just saw that you are mentionning the exact same reference in the comments, sorry. I guess that what you are looking for is Lemma 5 in the Ball-Carlen-Lieb paper.

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    $\begingroup$ I should have included this in my original question (my bad) but this is exactly the paper I'm referring to as proving the case of $1\leq p\leq 2$ but not the other case. I have also seen this work referenced to prove both cases but I just don't see it. $\endgroup$
    – Florian Ente
    Commented Jan 7, 2021 at 17:10
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    $\begingroup$ @ludoer Sorry, I read your question a bit too fast. Does the edit answer your question ? $\endgroup$
    – Mikael de la Salle
    Commented Jan 7, 2021 at 17:11
  • $\begingroup$ Just for clarification: Does the proof for theorem 1 imply what I've referred to as the opposite inequality (so the case $1\leq p \leq 2$) and Lemma 5 then says that the two cases are equivalent (for any normed space)? $\endgroup$
    – Florian Ente
    Commented Jan 7, 2021 at 17:21
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    $\begingroup$ @ludoer Yes, you are perfectly right. (Lemma 5 relates the convexity properties of $X$ with the smothness properties of its dual, and here we apply it to $S^p$ and $S^{p'}$). $\endgroup$
    – Mikael de la Salle
    Commented Jan 7, 2021 at 19:34
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    $\begingroup$ Yes, $S^{p'}$ is the dual of $S^p$. This follows from Hölder's inequality for matrices, see for example the survey by Pisier-Xu in my answer. $\endgroup$
    – Mikael de la Salle
    Commented Jan 7, 2021 at 22:05

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