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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 spectrum to be real whereas the latter helps in fixing a definition for inner-product of the eigenstates. Here we separate out adjointness of an operator from its Hermitian-adjointness. It turns out that a Hamiltonian possessing real spectrum is first pseudo-real, further it could be Hermitian, PT-symmetric ordoi:10.1088/0305-4470/36/41/005 fatcat:4rwyperklrgupo3qazvjxg2p6u