Can the Floquet theorem be generalized to Hilbert spaces? I think the generalization would look something like this:
Consider a dynamical system $\dot{x}(t)=A(t)x(t)$, where $A(t)$ is a family of operators such that $A(t)=A(t+T)$ for all $t$. A fundamental matrix solution is a family of operators $\phi(t)$ such that for initial condition $x(0)=x_0$, we have $$x(t)=\phi(t)\phi^{-1}(0)x_0.$$
Now the first part of theorem would state that for all fundamental matrix solutions $\phi$, we have
$$\phi(t+T)=\phi(t)\phi^{-1}(0)\phi(T).$$
Secondly, for each operator $B$ such that $e^{TB}=\phi^{-1}(0)\phi(T)$, there is a periodic (period $T$) family of operators $P(t)$ such that
$$\phi(t)=P(t)e^{tB}$$ for all $t\in\mathbb R$.
The last part of the theorem would say that for any orthogonal basis $(f_n)_n$ of the Hilbert space, there exists a real operator $R$ and a real family of operators $Q(t)$ (period $2T$) such that
$$\phi(t)=Q(t)e^{tR}$$ for all $t\in\mathbb R$.
Here a real operator $R$ means that $R(f_n)=\sum_{m}R_{mn}f_m$ where $R_{mn}\in\mathbb R$.