This problem arises when minimizing the operator equation $X P X^* + X Q + R$ over positive $X$ with respect to the positive cone on a Hilbert space $\mathcal{H}$.

The (reduced) task:

Given $P$ and $Q$ are positive (bounded) linear operators on $\mathcal{H}$, respectively, find a positive $X\in\mathcal{L}_+(\mathcal{H})$ so that $$H P X + X P H + HQ = 0$$ or equivalently $$H P X + X (HP)^* + HQ = 0$$ for all (bounded) operators $H\in\mathcal{L}_+(\mathcal{H})$.

Note that this implies $Q$ must commute with all positive $H$, so we might assume $Q=aI$ for some $a$ in the base field (if $H$ was allowed to be any operator this would be necessary. Since $H$ must be positive, are multiples of the identity still the only operators that always commute?)

My questions:

• under what (nontrivial) conditions does the above equation have a closed form solution?
• this seems related to the Lyapunov equation $A^*X + XA = W$ (if we let $H=I$). Is there some significance to this?