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Let $A $ and $ B $ be two self-adjoint and positive operators on a Hilbert space, and let $ 0 < b' \leq 2 \leq b $ and $ q > 1 $. Then, there exist constants ( C, C' > 0 ) such that:

$$ (A^{b'/2} B A^{b'/2})^{q/2} \leq C (A^{b/2} B A^{b/2})^{q/2} + C' I. $$ Thank you in advance

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    $\begingroup$ What to you assume about the dependence of the constants on all the parameters? If $C'$ can depend on $A$ and $B$, then one can do it even for $C=0$, simply because the left-hand side is a bounded self-adjoint operator. On the other hand, if the inequality works with some $C, C'$ independent of $A$ and $B$, then it also works with $C' = 0$, which can be seen by rescaling $B$. $\endgroup$ Commented Mar 24 at 11:01

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