ExactPT-symmetry is equivalent to Hermiticity

We show that a quantum system possessing an exact antilinear symmetry, in particular PT -symmetry, is equivalent to a quantum system having a Hermitian Hamiltonian. We construct the unitary operator relating an arbitrary non-Hermitian Hamiltonian with exact PT -symmetry to a Hermitian Hamiltonian. We apply our general results to PT -symmetry in finite dimensions and give the explicit form of the above-mentioned unitary operator and Hermitian Hamiltonian in two dimensions. Our findings lead to the conjecture that non-Hermitian CPT -symmetric field theories are equivalent to certain nonlocal Hermitian field theories.