  We in this paper demonstrate that the $PT$-symmetric non-Hermitian
Hamiltonian for a periodically driven system can be generated from a kernel
Hamiltonian by a generalized gauge transformation. The kernel Hamiltonian is
Hermitian and static, while the time-dependent transformation operator has to
be $PT$ symmetric and non-unitary in general. Biorthogonal sets of eigenstates
appear necessarily as a consequence of non-Hermitian Hamiltonian. We obtain
analytically the wave functions and associated non-adiabatic Berry phase
$\gamma_{n}$ for the $n$th eigenstate. The classical version of the
non-Hermitian Hamiltonian becomes a complex function of canonical variables and
time. The corresponding kernel Hamiltonian is derived with $PT$ symmetric
canonical-variable transfer in the classical gauge transformation. Moreover,
with the change of position-momentum to angle-action variables it is revealed
that the non-adiabatic Hannay's angle $\Delta \theta_{H}$ and Berry phase
satisfy precisely the quantum-classical correspondence,$\gamma_{n}=$
$(n+1/2)\Delta \theta_{H}$.
