Pseudo-reality and pseudo-adjointness of Hamiltonians

We define pseudo-reality and pseudo-adjointness of a Hamiltonian, H , as ρ H ρ −1 = H * and μ H μ −1 = H ', respectively. We prove that the former yields the necessary condition for a spectrum to be real whereas the latter helps in fixing a definition for the inner-product of the eigenstates. Here we separate out the adjointness of an operator from its Hermitian adjointness. It turns out that a Hamiltonian possessing a real spectrum is first pseudo-real, further it could be Hermitian, PT -symmetric or pseudo-Hermitian.