Let $A$ be some matrix and let $C_{A}=\{X| A^{T}X+XA>0\}$ be the cone of the Lyapunov solutions of the matrix $A$. Let also $P$ be the cone of positive definite matrices. If the Lyapunov operator $L_{A}$ is invertible then the dual cone is $C_{A}^{*}=-L_{A}P^{*}=\{-(AX+XA^{T})|X \geq 0\}$.
My question is: does this remain remain true when $L_{A}$ is not invertible?
