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
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.