The complex-conjugated loop $-C\,S_\infty^{\mathsf T}C$ is a nondegenerate symmetric loop when $S_\infty$ is

Let $S_\infty:\mathbb R\to\mathbb R^{2n\times2n}$ be a continuous $1$-periodic loop of symmetric matrices whose asymptotic operator $A=-J_0\partial_t-S_\infty(t)$ is nondegenerate. 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 $\bar S_\infty$ is again a continuous $1$-periodic loop of symmetric matrices, and its asymptotic operator $-J_0\partial_t-\bar S_\infty(t)$ is nondegenerate.

The loop $\bar S_\infty$ is the asymptotic data of the conjugated formal adjoint $-C\,D_S^*\,C=D_{\bar S}$ of a Cauchy–Riemann operator $D_S=\partial_s+J_0\partial_t+S$ (Wendl, Lecture 4). The statement lets results proved under nondegeneracy hypotheses be applied to that adjoint operator.

Formalization Note. The loop hypothesis is IsSymLoop (continuous, $1$-periodic, symmetric values). Nondegeneracy is IsNondegenerate: $\det(1-\Psi(1))\ne0$ for every fundamental solution $\Psi'=J_0S_\infty\Psi$, $\Psi(0)=1$. $C$ is conjMat n, and $J_0$ is Matrix.J.