Complex conjugation $S\mapsto-C\,S^{\mathsf T}C$ preserves asymptotic convergence of Cauchy–Riemann coefficients

Let $S:\mathbb R\times\mathbb R\to\mathbb R^{2n\times2n}$ be continuous and $1$-periodic in $t$, and let $S^\pm$ be loops such that $S(s,\cdot)\to S^\pm$ entrywise and uniformly in $t$ as $s\to\pm\infty$. Let $C=\begin{pmatrix}\mathrm{Id}&0\\0&-\mathrm{Id}\end{pmatrix}$ be complex conjugation on $\mathbb C^n=\mathbb R^{2n}$, and set

Then $\bar S$ is continuous and $1$-periodic in $t$, and $\bar S(s,\cdot)\to\bar S^\pm$ entrywise and uniformly in $t$ as $s\to\pm\infty$.

The coefficient $\bar S$ is that of the conjugated formal adjoint $D_{\bar S}=-C\,D_S^*\,C$ of $D_S=\partial_s+J_0\partial_t+S$ (Wendl, Lecture 4). The statement transfers the asymptotic hypothesis from $D_S$ to $D_{\bar S}$.

Formalization Note. Asymptotic convergence is IsAsymptoticTo S Splus Sminus: continuity, $t$-periodicity, and for each $\varepsilon>0$ an $R$ with $|S(s,t)_{ij}-S^\pm(t)_{ij}|\le\varepsilon$ for $\pm s\ge R$. $C$ is conjMat n. No symmetry of $S^\pm$ is assumed.