  We study in this paper the time evolution of $PT$-symmetric non-Hermitian
Hamiltonian consisting of periodically driven $SU(1,1)$ generators. A
non-Hermitian invariant operator is adopted to solve the Schr\"{o}dinger
equation, since the time-dependent Hamiltonian is no longer a conserved
quantity. We propose a scheme to construct the non-Hermitian invariant with a
$PT$-symmetric but non-unitary transformation operator. The eigenstates of
invariant and its complex conjugate form a bi-orthogonal basis to formulate the
exact solution. We obtain the non-adiabatic Berry phase, which reduces to the
adiabatic one in the slow time-variation limit. A non-unitary time-evolution
operator is found analytically. As an consequence of the non-unitarity the ket
($|\psi (t)\rangle $) and bra ($\langle \psi (t)|$) states are not normalized
each other. While the inner product of two states can be evaluated with the
help of a metric operator. It is shown explicitly that the model can be
realized by a periodically driven oscillator.
