  The Heisenberg picture for non-Hermitian but $\eta$-pseudo-Hermitian
Hamiltonian systems is suggested. If a non-Hermitian but
$\eta$-pseudo-Hermitian Hamiltonian leads to real second order equations of
motion, though their first order Heisenberg equations of motion are complex, we
can construct a Hermitian counterpart that gives the same second order
equations of motion. In terms of a similarity transformation we verify the
iso-spectral property of the Hermitian and non-Hermitian Hamiltonians and
obtain the related eigenfunctions. This feature can be used to determine real
eigenvalues for such non-Hermitian Hamiltonian systems. As an application, two
new non-Hermitian Hamiltonians are constructed and investigated, where one is
non-Hermitian and non-PT-symmetric and the other is non-Hermitian but
PT-symmetric. Moreover, the complementarity and compatibility between our
treatment and the PT symmetry are discussed.
