The asymptotic operator of a symmetric loop is self-adjoint

Statement. For every $n \in \mathbb{N}$ (including $n = 0$) and every continuous map $S : S^1 \to \mathrm{Mat}_n$ such that $S(t)^{\mathsf T} = S(t)$ for every $t \in S^1$, the operator $\mathcal A_S$ (defined below) equals its own adjoint: $\mathcal A_S^\dagger = \mathcal A_S$ as partially defined operators, i.e. they have the same domain and agree on it.

The adjoint used. For a partially defined real-linear operator $A$ on $L^2_n$:

• $\operatorname{dom} A^\dagger$ is the set of $y \in L^2_n$ such that the linear functional $x \mapsto \langle y, Ax\rangle_{L^2}$ on $\operatorname{dom} A$ is continuous for the $L^2$ norm restricted to $\operatorname{dom} A$;
• if $\operatorname{dom} A$ is dense in $L^2_n$, then $A^\dagger y$ is the unique $w \in L^2_n$ with $\langle w, x\rangle = \langle y, Ax\rangle$ for all $x \in \operatorname{dom} A$; if $\operatorname{dom} A$ is not dense, $A^\dagger$ is (by convention) the zero map on that domain.

So the claim is: $y \in \operatorname{dom}\mathcal A_S$ iff $x \mapsto \langle y, \mathcal A_S x\rangle$ is $L^2$-continuous on $\operatorname{dom}\mathcal A_S$, and for such $y$, $\mathcal A_S^\dagger y = \mathcal A_S y$ (with $\mathcal A_S^\dagger$ as just described).

Meaning of the other symbols.

• $S^1 = \mathbb{R}/\mathbb{Z}$ with its Haar probability measure $\lambda$; $\mathrm{Mat}_n$ is the space of real $2n\times 2n$ matrices indexed by $\{1,\dots,n\}\sqcup\{1,\dots,n\}$; $J_0 = \begin{pmatrix} 0 & -I_n \\ I_n & 0\end{pmatrix}$.
• $V_n = \mathbb{R}^{2n}$ with the Euclidean inner product, and $L^2_n = L^2(S^1,\lambda;V_n)$, the real Hilbert space of a.e.-classes of square-integrable maps with $\langle f, g\rangle_{L^2} = \int_{S^1}\langle f(t), g(t)\rangle\, d\lambda(t)$ and norm $\|f\| = (\int |f|^2\, d\lambda)^{1/2}$. Pointwise values of an element of $L^2_n$ are those of an arbitrary measurable representative.
• Fourier coefficients: $\hat f_k = \int_{S^1} e^{-2\pi i k t} f(t)\, d\lambda(t) \in \mathbb{C}^{2n}$ and $\widehat{(Sf)}_k = \int_{S^1} e^{-2\pi i k t} S(t) f(t)\, d\lambda(t)$, $k \in \mathbb{Z}$ (real entries viewed in $\mathbb{C}$; Bochner integral, value $0$ if not integrable).
• $f$ is "in $W^{1,2}$" iff $\sum_{k\in\mathbb{Z}} (1+k^2)\sum_i |(\hat f_k)_i|^2$ is summable, $|\cdot|$ the complex modulus, $i$ over all $2n$ coordinates.
• $\Gamma_S$ is the set of pairs $(f, g)\in L^2_n\times L^2_n$ with $f$ in $W^{1,2}$ and $\hat g_k = -(2\pi k\, i) J_0 \hat f_k - \widehat{(Sf)}_k$ for all $k \in \mathbb{Z}$.
• $\mathcal A_S$ is defined by cases: if some partially defined real-linear operator on $L^2_n$ has graph exactly $\Gamma_S$, then $\mathcal A_S$ is such an operator (picked by choice); otherwise $\mathcal A_S$ is the zero operator with domain all of $L^2_n$ (which is also self-adjoint in the above sense). The theorem is about this $\mathcal A_S$ whichever case occurs.

For $n = 0$, $L^2_0 = \{0\}$ and the claim concerns operators on the zero space.