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$\DeclareMathOperator\tr{tr}$Suppose we have positive semidefinite matrices $A_1, \dotsc, A_n$ and $B_1, \dotsc, B_n$ of the same dimension. Do we have a Hölder inequality for the trace of the following form: $$\tr\Big(\sum\limits_{j=1}^n A_j B_j\Big) \leq \tr\Big[\Big(\sum\limits_{j=1}^n A_j^p\Big)^{\frac{1}{p}} \Big(\sum\limits_{j=1}^n B_j^q\Big)^{\frac{1}{q}}\Big]$$

for $1 = \frac{1}{p} + \frac{1}{q}$? What if $p = q = 2$?

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    $\begingroup$ No proof for this one, but for the record, two somewhat similar inequalities with a proof: $${\rm tr}\,\sum\limits_{j=1}^n A_j B_j \leq {\rm tr}\,\sum\limits_{j=1}^n\Big( A_j^p/p+B_j^q/q\Big)$$ $${\rm tr}\,\sum\limits_{j=1}^n A_j B_j \leq\Big({\rm tr}\,\sum_{j=1}^nA_j^p\Big)^{1/p}\Big({\rm tr}\,\sum_{j=1}^nB_j^q\Big)^{1/q},$$ $\endgroup$
    – Carlo Beenakker
    Commented Mar 23, 2022 at 14:16

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