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

The (reduced) task:

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

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 $H=I$) but more general in that the adjoint $X^*$ appears. Is there some significance to this? Does this adjoint make the system much harder to solve?