Spectrum of the unperturbed operator $-J_0\partial_t$

Setting. Let $n \in \mathbb{N}$ be arbitrary ($n = 0$ is allowed). Let $I = \{1,\dots,n\} \sqcup \{1,\dots,n\}$ (a first and a second copy), $V_n = \mathbb{R}^I \cong \mathbb{R}^{2n}$ with the standard Euclidean inner product, and $\mathrm{Mat}_n$ the real $I \times I$ (i.e. $2n \times 2n$) matrices. In block form with respect to the two copies, $J_0 = \begin{pmatrix} 0 & -\mathrm{Id}_n \\ \mathrm{Id}_n & 0 \end{pmatrix}$. Let $\mathbb{T} = \mathbb{R}/\mathbb{Z}$ with its Haar probability measure $dt$ (total mass $1$), and let $H = L^2(\mathbb{T}, V_n)$ be the real Hilbert space of a.e.-classes of square-integrable functions, with $\langle f, g \rangle = \int_{\mathbb{T}} \langle f(t), g(t) \rangle \, dt$. For $n = 0$, $V_0 = \{0\}$ and $H = \{0\}$.

For $f \in H$ (any representative) and $k \in \mathbb{Z}$, the Fourier coefficient is the vector $\hat f_k = \int_{\mathbb{T}} e^{-2\pi i k t} f(t)\, dt \in \mathbb{C}^I$, where $f(t)$ is regarded as a complex vector componentwise (a Bochner integral). Say $f \in W^{1,2}$ if the family $\big((1 + k^2) \sum_{i \in I} |(\hat f_k)_i|^2\big)_{k \in \mathbb{Z}}$ is summable.

Here the loop is $S = 0$, the constant zero matrix-valued map on $\mathbb{T}$; its term $\widehat{(Sf)}_k = \int_{\mathbb{T}} e^{-2\pi i k t}\, 0 \cdot f(t)\, dt$ is $0$. So $G_0 \subseteq H \times H$ is the set of pairs $(f, g)$ with $f \in W^{1,2}$ and

The operator $A_0$ is defined as follows: if there exists a partially defined linear map on $H$ (a linear subspace $D \subseteq H$ and a linear map $T : D \to H$) whose graph $\{(x, Tx) : x \in D\}$ equals $G_0$, then $A_0$ is such a map, chosen by the axiom of choice (its domain $D(A_0)$ is then $\{f : \exists g,\ (f,g) \in G_0\}$ and $A_0 f$ is the corresponding $g$); if no such map exists, $A_0$ is the zero map with domain all of $H$. The statement does not itself assert which case occurs.

For $\mu \in \mathbb{R}$, the eigenspace $E_\mu(A_0)$ is the real linear subspace $\{ f \in D(A_0) : A_0 f = \mu f \}$ of $D(A_0)$.

Statement. For every $n \in \mathbb{N}$ and every $\mu \in \mathbb{R}$, both of the following hold:

1. If $\mu = 2\pi k$ for some integer $k \in \mathbb{Z}$, then there exists a real-linear isomorphism between $E_\mu(A_0)$ and the space $\mathbb{R}^I \cong \mathbb{R}^{2n}$ of real functions on $I$; i.e. $E_\mu(A_0)$ is a real vector space of dimension exactly $2n$.
2. If $\mu \neq 2\pi k$ for every $k \in \mathbb{Z}$, then $E_\mu(A_0) = \{0\}$: the only $f \in D(A_0)$ with $A_0 f = \mu f$ is $f = 0$.

No symmetry or other hypotheses are involved (the loop is fixed to be $0$). For $n = 0$, $H = \{0\}$, every eigenspace is $\{0\}$, and part 1 asks for an isomorphism with the zero-dimensional space $\mathbb{R}^0$.