  We investigate some questions on the construction of $\eta$ operators for
pseudo-Hermitian Hamiltonians. We give a sufficient condition which can be
exploited to systematically generate a sequence of $\eta$ operators starting
from a known one, thereby proving the non-uniqueness of $\eta$ for a particular
pseudo-Hermitian Hamiltonian. We also study perturbed Hamiltonians for which
$\eta$'s corresponding to the original Hamiltonian still work.
