Bounds on Singular Values or Inverse of Sylvester Operator

Question

For real square matrices $A$ and $B$ consider the Sylvester operator $S(X) = AX - XB$. The eigenvalues of the operator are all differences between the eigenvalues of $A$ and $B$, so $S$ is invertible if the sets of eigenvalues of $A$ and $B$ are disjoint. Intuitively then, there should be a lower bound on the minimum singular value of $S$ in terms of the minimum distance between any two eigenvalues of $A$ and $B$, but I can find no such bound. Is any such simple bound known?