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Do Hilbert-Schmidt operators which are also trace class, satisfy Hölder's inequality? That is, we have two Hilbert-Schmidt operators $A$ and $B$. Is the following true? $$\langle A, B \rangle \leq \lVert A \rVert_p \lVert B \rVert_q $$

when $1/p + 1/q = 1$. Specifically, is it true for $p=1$, $q=\infty$?

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    $\begingroup$ Yes, this is standard stuff. Note that any trace class operator belongs to all the Schatten $p$-spaces. We do have a Holder inequality. In fact, if $A$ is trace class and $B$ is any bounded operator then $|tr(AB^*)| \leq \|A\|_1 \|B\|$. You should be able to find this in most textbooks. $\endgroup$
    – Nik Weaver
    Commented Apr 13, 2018 at 5:53
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    $\begingroup$ (I assume you mean $|\langle A, B\rangle|$, not $\langle A,B\rangle$.) $\endgroup$
    – Nik Weaver
    Commented Apr 13, 2018 at 5:54
  • $\begingroup$ Thanks! Yes I mean $|<A,B>|$. Can you point me to a reference (textbook, paper)? $\endgroup$
    – Another Grad student
    Commented Apr 13, 2018 at 15:21
  • $\begingroup$ Probably in Dunford-Schwartz, likely in Conway. The $p=1$, $q = \infty$ inequality is in chapter 6 of my book Mathematical Quantization. $\endgroup$
    – Nik Weaver
    Commented Apr 13, 2018 at 19:55

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