The kernel of the asymptotic operator is isomorphic to $\ker(\Psi(1)-\mathrm{Id})$

Matrices. Let $n \ge 0$ be a natural number ($n = 0$ is allowed). Write $\mathrm{Mat}_n$ for the real $2n \times 2n$ matrices, rows and columns indexed by two disjoint copies of $\{0,\dots,n-1\}$, and

$$J_0 = \begin{pmatrix} 0 & -I_n \\ I_n & 0 \end{pmatrix}.$$

Let $S^1 = \mathbb{R}/\mathbb{Z}$; for $t \in \mathbb{R}$, $S(t)$ means $S$ at the class $t \bmod 1$.

The space $L^2$. $\mathbb{R}^{2n}$ carries its Euclidean inner product. $L^2 = L^2(S^1, \mathbb{R}^{2n})$ is the real Hilbert space of (classes modulo almost-everywhere equality of) measurable square-integrable functions $f : \mathbb{R}/\mathbb{Z} \to \mathbb{R}^{2n}$, with respect to the Haar probability measure $dt$ on $\mathbb{R}/\mathbb{Z}$ (total mass $1$). When $n = 0$, $L^2$ is the zero space.

Fourier coefficients. For $f \in L^2$ (evaluated through any representative; the integral does not depend on the choice) and $k \in \mathbb{Z}$, view $f(t)$ as a vector in $\mathbb{C}^{2n}$ (real entries) and set

$$\hat f_k = \int_{\mathbb{R}/\mathbb{Z}} e^{-2\pi i k t}\, f(t)\, dt \in \mathbb{C}^{2n},$$

and similarly

$$(Sf)^{\wedge}_k = \int_{\mathbb{R}/\mathbb{Z}} e^{-2\pi i k t}\, S(t) f(t)\, dt \in \mathbb{C}^{2n},$$

where $S(t)$ acts on the complexified vector by matrix–vector multiplication. (These are Bochner integrals; by the general convention an integral of a non-integrable function is $0$.)

$W^{1,2}$ condition. $f \in L^2$ is said to be in $W^{1,2}$ if the family

$$k \mapsto (1 + k^2) \sum_{i} \big|(\hat f_k)_i\big|^2 \qquad (k \in \mathbb{Z})$$

is summable over $\mathbb{Z}$, i.e. $\sum_{k\in\mathbb{Z}} (1+k^2)\,|\hat f_k|^2 < \infty$ (here $|\cdot|$ is the complex modulus and $i$ runs over the $2n$ coordinates).

Graph relation. For a continuous $S : \mathbb{R}/\mathbb{Z} \to \mathrm{Mat}_n$, let $G_S \subseteq L^2 \times L^2$ be the set of pairs $(f, g)$ such that $f \in W^{1,2}$ and, for every $k \in \mathbb{Z}$,

$$\hat g_k = -(2\pi k\, i)\, J_0 \hat f_k - (Sf)^{\wedge}_k ,$$

with $J_0$ acting on $\mathbb{C}^{2n}$ by its (real) entries.

The operator $A_S$. If there exists a partially defined $\mathbb{R}$-linear map $A$ on $L^2$ (a linear subspace $D(A) \subseteq L^2$ and a linear map $A : D(A) \to L^2$) whose graph $\{(x, Ax) : x \in D(A)\}$ equals $G_S$ as a subset of $L^2 \times L^2$, then $A_S$ is one such map, chosen by the axiom of choice. Otherwise, $A_S$ is by fallback the zero map with domain all of $L^2$ ($D(A_S) = L^2$, $A_S x = 0$ for all $x$).

Eigenspace. For a partially defined map $A$ and $\mu \in \mathbb{R}$, $\mathrm{eigenspace}(A, \mu)$ is the real vector space $\{x \in D(A) : A x - \mu x = 0\}$, a linear subspace of $D(A)$. Here $\mu = 0$, so it is $\{x \in D(A_S) : A_S x = 0\}$.

Hypotheses.

• $S : \mathbb{R}/\mathbb{Z} \to \mathrm{Mat}_n$ is continuous, and $S(t)^{\mathsf T} = S(t)$ for every $t$.
• $\Psi : \mathbb{R} \to \mathrm{Mat}_n$ is a fundamental solution of $S$: $\Psi(0) = I_{2n}$, and for every $t \in \mathbb{R}$, $\Psi$ is differentiable at $t$ (entrywise topology on $\mathrm{Mat}_n$) with $\Psi'(t) = J_0\, S(t)\, \Psi(t)$.

Conclusion. There exists an $\mathbb{R}$-linear isomorphism (bijective linear map; no continuity or other property is asserted)

$$\{x \in D(A_S) : A_S x = 0\} \;\cong\; \ker\big(\Psi(1) - I_{2n}\big) = \{v \in \mathbb{R}^{2n} : \Psi(1) v = v\},$$

where the right side is a linear subspace of $\mathbb{R}^{2n}$ (vectors indexed by the $2n$ coordinates, $\Psi(1) - I$ acting by matrix–vector multiplication). The statement asserts only that such an isomorphism exists; it does not specify which map.