We show that a diagonalizable non-Hermitian Hamiltonian H is pseudo-Hermitian if and
only if it has an antilinear symmetry, ie, a symmetry generated by an invertible antilinear
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 antilinear 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 …