Meaning of time evolution of Floquet matrix

Consider the time-periodical Hamiltonian $H(t)=H(t+T)$. In the Floquet theorem, the Schrödinger equation has a solution of the form
begin{align}
|psi_alpha(t)rangle=e^{-iepsilon_alpha t}|phi_alpha(t)rangle,
end{align}
where $|phi_alpha(t)rangle=|phi_alpha(t+T)rangle$. This Floquet eigenstates satisfy
begin{align}
left(H(t)-i frac{partial}{partial t}right)left|phi_{alpha}(t)rightrangle=epsilon_{alpha}left|phi_{alpha}(t)rightrangle.
end{align}
After the Fourier transformation, we get time-independent equation
begin{align}
H_F|Phiranglerangle=epsilon_alpha |Phiranglerangle,
end{align}
where the state $|Phiranglerangle$ belongs to the extended Hilbert space $mathcal{L}^2(0,T)otimes mathcal{H}$ whose ONB is written as ${|nrangleotimes|arangle}_{n,a}$.

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My question is as follows: I think the eigenstates and eigenvalue of $H_F$ are important because we can extract the information of the dynamics. However, is there any meaning of $exp(-iH_Ft)$? I cannot understand the meaning because the dynamics which is described by
begin{align}
ifrac{partial}{partial t}|Psi(t)ranglerangle=H_F|Psi(t)ranglerangle
end{align}
does not include any information of $|psi_alpha(t)rangle$.

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Sometimes people consider the Green’s function of $H_F$. However, if there is not any meaning of $exp(-iH_Ft)$, I do not understang the reason why people consider the Green’s funtion of $H_F$.