  We study a diagonalizable Hamiltonian that is not at first hermitian.
Requirement that a measurement shall not change one Hamiltonian eigenstate into
another one with a different eigenvalue imposes that an inner product must be
defined so as to make the Hamiltonian normal with regard to it. After a long
time development with the non-hermitian Hamiltonian, only a subspace of
possible states will effectively survive. On this subspace the effect of the
anti-hermitian part of the Hamiltonian is suppressed, and the Hamiltonian
becomes hermitian. Thus hermiticity emerges automatically, and we have no
reason to maintain that at the fundamental level the Hamiltonian should be
hermitian. If the Hamiltonian is given in a local form, a conserved probability
current density can be constructed with two kinds of wave functions. We also
point out a possible misestimation of a past state by extrapolating back in
time with the hermitian Hamiltonian. It is a seeming past state, not a true
one.
