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?

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?

Update:

I tried the naive solution on $\mathbb{Z}^n,\mathbb{R}^n,$ and $\mathbb{C}^n$ of setting $H=I$, solving the resulting Lyapunov equation for $X$, and then plugging $X$ back in to the derivative. Interestingly enough, in all my experiments this naive $X$ satisfies $HPX + XPH + HQ = 0$ for all symmetric $H$. Is this some weird scale-invariance for the Lyapunov Equation?

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?

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?

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 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?

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?