I'm trying to answer the following question: Given two positive self adjoint operators $\mathcal{A}$ and $\mathcal{P}$ on a Hilbert space, is the following composition: $\mathcal{AP}+\mathcal{PA}$ also positive?
One possible condition under which this is true is when $\mathcal{AP}=\mathcal{PA}$. Thus, for this case, we assume that the product of the operators do not commute.
What might be the conditions, in addition to the one stated previously, under which the question has an affirmative answer?
Regards,
Observe that \begin{align*} \langle (\mathcal A \mathcal P+\mathcal P\mathcal A)x,x\rangle&=\langle \mathcal A \mathcal Px,x\rangle+\langle\mathcal P\mathcal Ax,x\rangle\\ &=\langle \mathcal Px,\mathcal Ax\rangle+\langle\mathcal Ax,\mathcal Px\rangle\\ &=2\mbox{Re }\langle\mathcal Px,\mathcal Ax\rangle=2\mbox{Re }\langle\mathcal A\mathcal Px,x\rangle. \end{align*} Thus, $\langle (\mathcal A \mathcal P+\mathcal P\mathcal A)x,x\rangle$ is positive if and only if the real part of $\langle\mathcal A\mathcal Px,x\rangle$ lies in the right half plane. For accretive operators, do a google search or check out the book by Konrad Schmüdgen. They are related to generators of contraction semigroups.