  It is shown that if the C operator for a PT-symmetric Hamiltonian with simple
eigenvalues is not unique, then it is unbounded. Apart from the special cases
of finite-matrix Hamiltonians and Hamiltonians generated by differential
expressions with PT-symmetric point interactions, the usual situation is that
the C operator is unbounded. The fact that the C operator is unbounded is
significant because, while there is a formal equivalence between a PT-symmetric
Hamiltonian and a conventionally Hermitian Hamiltonian in the sense that the
two Hamiltonians are isospectral, the Hilbert spaces are inequivalent. This is
so because the mapping from one Hilbert space to the other is unbounded. This
shows that PT-symmetric quantum theories are mathematically distinct from
conventional Hermitian quantum theories.
