An asymptotic operator is nondegenerate exactly when its fundamental solution lies in $\mathrm{SP}(n)$

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

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

A matrix $A$ is symplectic if $A\, J_0\, A^{\mathsf T} = J_0$. 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. $\mathrm{eigenspace}(A_S, 0)$ is the real vector space $\{x \in D(A_S) : A_S x - 0\cdot x = 0\} = \{x \in D(A_S) : A_S x = 0\}$, a linear subspace of $D(A_S)$. "$\mathrm{eigenspace}(A_S, 0) = 0$" means this subspace consists of the zero vector only.

The set $\mathrm{SP}(n)$. $\mathrm{SP}(n)$ is the set of continuous maps $\psi : [0,1] \to \mathrm{Mat}_n$ such that $\psi(t)$ is symplectic for every $t \in [0,1]$, $\psi(0) = I_{2n}$, and $\det\big(I_{2n} - \psi(1)\big) \neq 0$. (For $n = 0$, the determinant of the empty matrix is $1$, so this last condition holds automatically.)

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 $\Psi'(t) = J_0\, S(t)\, \Psi(t)$.
• $\psi : [0,1] \to \mathrm{Mat}_n$ is a continuous map with $\psi(t) = \Psi(t)$ for every $t \in [0,1]$ (i.e. $\psi$ is the restriction of $\Psi$ to $[0,1]$, supplied as a separate continuous map).

Conclusion.

$$\{x \in D(A_S) : A_S x = 0\} = \{0\} \quad\Longleftrightarrow\quad \psi \in \mathrm{SP}(n),$$

i.e. the zero-eigenspace of $A_S$ is trivial if and only if $\psi(t)$ is symplectic for all $t \in [0,1]$, $\psi(0) = I_{2n}$, and $\det(I_{2n} - \psi(1)) \neq 0$.