Multiplication by a continuous matrix loop preserves $L^2(S^1,\mathbb{R}^{2n})$

Let $S^1=\mathbb R/\mathbb Z$ with its Haar probability measure and let $L^2_n=L^2(S^1,\mathbb R^{2n})$. For $f\in L^2_n$ and $k\in\mathbb Z$ let $\hat f_k=\int_{S^1}e^{-2\pi ikt}f(t)\,dt\in\mathbb C^{2n}$ be its $k$-th Fourier coefficient, computed componentwise after viewing the real entries of $f$ as complex numbers.

Let $S:S^1\to\mathbb R^{2n\times2n}$ be continuous (no symmetry is assumed) and $f\in L^2_n$. Then there is $m\in L^2_n$ with $\hat m_k=(Sf)^\wedge_k$ for every $k\in\mathbb Z$, where $(Sf)^\wedge_k=\int_{S^1}e^{-2\pi ikt}S(t)f(t)\,dt$.

In other words, multiplication by a continuous matrix loop maps $L^2(S^1,\mathbb R^{2n})$ into itself. This is the zeroth-order term of the asymptotic operator $A_S=-J_0\partial_t-S$.

Formalization note: L2 n, coeff and coeffMul come from the definition module AsymptoticOperator_Setting; L2 n is Lp (EuclideanSpace ℝ (Fin n ⊕ Fin n)) 2 over UnitAddCircle with AddCircle.haarAddCircle, coeff m k is $\hat m_k$ and coeffMul S f k is $(Sf)^\wedge_k$, with $S(t)$ applied to the complexified vector $f(t)$.