Pseudo-Hermiticity versus PT Symmetry III: Equivalence of pseudo-Hermiticity and the presence of anti-linear symmetries

We show that a (non-Hermitian) Hamiltonian H admitting a complete biorthonormal set of eigenvectors is pseudo-Hermitian if and only if it has an anti-linear symmetry, i.e., a symmetry generated by an anti-linear operator. This implies that the eigenvalues of H are real or come in complex conjugate pairs if and only if H possesses such a symmetry. In particular, the reality of the spectrum of H implies the presence of an anti-linear symmetry. We further show that the spectrum of H is real if and only if there is a positive-definite inner-product on the Hilbert space with respect to which H is Hermitian or alternatively there is a pseudo-canonical transformation of the Hilbert space that maps H into a Hermitian operator.