We define pseudo-reality and pseudo-adjointness of a Hamilt onian, H, as ρH ρ⁻¹ = H^∗ and μH 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 or pseudo-Hermitian.