Negative spectra do not imply uniform semidiscrete stability ($L_h=\left(\begin{array}{cc}-1& h^{-1}\\ 0& -1\end{array}\right)$)

Every eigenvalue of this nonnormal matrix is $-1$, but $e^{t{L}_h}=e^{-t}\left(\begin{array}{cc}1& t/h\\ 0& 1\end{array}\right)$. Applied to the second coordinate unit vector, its norm at $t=1$ is at least $1/(eh)$. Thus negative eigenvalues give no mesh-uniform semigroup bound by themselves. Normal matrices, uniformly conditioned eigenvector bases, or a direct dissipative energy estimate supply the missing control.