Bounds on Lyapunov Singular Values

Question

Let $A$ be a square $n \times n$ real matrix, and consider the Lyapunov operator $L(X)$ defined by $$ L(X) := A^T X A - X \:. $$ I am curious if there are any known lower bounds on the following quantity $$ \underline{\sigma} := \inf\{ \| L(X) \|_F : \|X\|_F = 1 \} \:, $$ in terms of some properties of the spectrum of the $A$ matrix. Using the identity $\mathrm{vec}(A X B) = (B^T \otimes A) \mathrm{vec}(X)$, we can see that this is equivalent to computing the minimum singular value of the matrix $$ A^T \otimes A^T - I \:. $$ Weyl's inequality gives us a bound $$ \sigma_{\min}(A^T \otimes A^T - I) \geq 1 - \sigma_{\max}(A)^2 \:. $$ But if $\sigma_{\max}(A) \geq 1$, then this bound is vacuous. I'm curious if we can say something under weaker conditions. Thanks!