$\mathcal{H}_\infty$ norm of a system is a lower bound of the $\mathcal{L}_1$ norm?

Question

For a stable causal SISO LTI system $G(s)$, let $H = \|G(s)\|_{\mathcal{H}_\infty}$, and let $\omega^*$ be the frequency at which this is achieved$^\dagger$. The output of the system to an input of $\cos(\omega^*t)$ then has magnitude $H$. Thus $\| G(s) \|_{\mathcal{L}_1} \geq H$.

However, this line of reasoning does not hold for MIMO systems since the output under $\cos(\omega^*t)$ need not have a magnitude of $H$. Is it still true that $\| G(s) \|_{\mathcal{L}_1} \geq H$ for MIMO systems? How else might we show this?

_{$^\dagger$If the norm is only reached asymptotically, take limits as necessary}

Answer 1

It turns out it is true. Just follow the same reasoning and apply the input along the (right) singular vector associated with the max singular value in the $\mathcal{H}_\infty$ norm. The output is then a phase-shifted sinusoid with magnitude $H$ and occurs along the (left) singular vector.