Given a matrix $A$, whose eigenvalues are $\lambda_i$ with $\Re(\lambda_i)\lt 0$, is there a matrix $H$ such that
$$HA + A^* H = -I$$
where $I$ is the identity matrix and $A^*$ is the adjoint matrix? If such a matrix $H$ exists, is it unique? Is it possible to write out an expression for $H$?
Please give small hints, but do not completely solve the problem for me.