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Suppose I have two $d \times d$ entire matrix functions $F, G$ defined on $\mathbb{C}$ with the the property that $\|FG^*\|_{L^\infty(\mathbb{C})} < \infty$. Can anything be said about $F$ and $G$, or even the product $FG$?

Trivially one has $\|\det FG\|_{L^\infty(\mathbb{C})} < \infty$ and so by Liouville's theorem $\det FG$ is constant, but I don't really see what is useful about this, other than either both $F$ and $G$ are invertible everywhere, or that for every $z \in \mathbb{C}$ either $F(z)$ or $G(z)$ (or both) are singular.

Any thoughts or places to look would be very helpful!

Note: for further context, this question immediately pops up when looking at bounded Toeplitz products $T_F T_{G^*}$ on the Segal Bargman/Fock space with entire symbols $F$ and $G$.

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    $\begingroup$ I'm not sure if this is what you're asking, but it's not possible to conclude that $F$, $G$, or $FG$ is bounded (constant). There is the trivial counterexample $$F=\begin{pmatrix} f & 0 \\ 0&0\end{pmatrix}, \quad G=\begin{pmatrix} 0 & g \\ 0&0 \end{pmatrix}. $$ $\endgroup$
    – Christian Remling
    Commented Jun 27, 2023 at 0:40
  • $\begingroup$ Christian, yes thanks for mentioning trivial counterexamples like this. $\endgroup$
    – Joshua Isralowitz
    Commented Jun 28, 2023 at 16:54

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