  We define pseudo-reality and pseudo-adjointness of a Hamiltonian, $H$, as
$\rho H \rho^{-1}=H^\ast$ and $\mu H \mu^{-1}=H^\prime$, respectively. We prove
that the former yields the {\it 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.

Quantum Physics