Given $P \geq 0$, I need to show that $2Tr(P^{5/2}) \leq Tr(P^3) + Tr(P^2)$. It's trivial to show that the RHS is the trace of a positive operator, but I'm at a loss on how to actually prove this inequality...
2026-04-02 09:44:14.1775123054
Help proving operator inequality
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We have $P^3 + P^2 - 2 P^{\frac{5}{2}} = (P^{\frac{3}{2}} - P)^{2} \geq 0$, so the trace is also non-negative, and this is the announced inequality.