Given $f\in \mathbb{H}^{1\times m}_{2}(U) $ and $g\in\mathbb{H}^{m\times m}_{2}(U)$.
Is it true that $fg\in \mathbb{H}^{1\times m}_{2}(U)$ ??
If not, what conditions needed so that it would be true!
*Is the problem related to the Smirnov maximum principle?
Where:
$\mathbb{H}^{n\times m}_{2}(U)$ is the Hardy space of $n\times m$ matrix valued functions with entries in the space $\mathbb{H}_{2}(U)$, in the upper half plane $U$, with norm $$ \|f\|_{\mathbb{H}_{2}^{n\times m}}^{2} = \sup_{y>0} \int_{-\infty}^{\infty} trace\{f^{*}(x+ iy)f(x+iy)\}\, \mathrm{d}x $$