I am looking for a proof of the following:$$ |||\frac{1}{2} AB + \frac{1}{2} BA||| \leq |||AB|||$$
For positive hermitian matrices A and B, and a unitarily invariant norm $ |||\cdot|||$.
2026-04-11 23:01:24.1775948484
Proof: $ |||\frac{1}{2} AB + \frac{1}{2} BA||| \leq |||AB|||$
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Since $\|\cdot\|$ is unitarily invariant, $\|M\|$ depends only on the singular values of $M$ and since $M^*$ has the same singular values as $M$, $\|M\|=\|M^*\|$. It remains to use the triangle inequality.