I am looking for an approximation result dealing with continuous functions of a real parameter with values in (some subset of) the unitary algebra. However, I wouldn't be surprised if the following statement — or an even more general version thereof — holds/can be found somewhere in the literature:
Let $\mathfrak g$ be a Lie subalgebra of $\mathfrak{gl}(n)$. Given $C\subseteq\mathfrak g$ convex & closed and $f:\mathbb R\to C$ continuous, for every $\varepsilon>0$ there exists $g:\mathbb C\to \mathfrak g$ holomorphic with $g(\mathbb R)\subseteq C$ such that $\|f-g|_{\mathbb R}\|_{\rm sup}<\varepsilon$.
While my problem deals with functions $f$ defined on an interval $I\subseteq\mathbb R$ this does not restrain anything so w.l.o.g. $I=\mathbb R$. Either way simply applying Carleman's original approximation theorem to the entries $f_{jk}(t):=e_j^Tf(t)e_k$ takes care of the special cases $\mathfrak g=\mathfrak{gl}(n)=C$ as well as $\mathfrak g=\mathfrak u(n)=C$ (e.g., after anti-symmetrizing the approximation function).
However, I'm interested the instance where the codomain $C$ of $f$ is a proper (closed, convex) subset of $\mathfrak u(n)$: more precisely, for me
$C=C_1\cap C_2$ is the intersection of the closed convex cone $C_1:=\{X\in\mathfrak u(n):iX\geq 0\}$ and the kernel $C_2:={\rm ker}(h)$ of an affine map $h:\mathfrak{u}(n)\to\mathfrak{u}(n)$ (or rather the restriction of an affine, Hermiticity-preserving map $\hat h:\mathfrak{gl}(n)\to\mathfrak{gl}(n)$).
Now if $C=C_1$, then I feel like Carleman's result may still do the job as non-negativity is not a restriction (at least if the co-domain is one-dimensional). However, intersecting $C_1$ with $C_2$ seems to obscure the connection to standard approximation results as this places constraints on the values of $f$ which cannot be expressed easily in terms of the components of $f$. Thanks in advance for any answer or comment!
