  In the recently quickly developing context of quantum mechanics of unitary
systems using a time-independent non-Hermitian Hamiltonian $H$ (having real
spectrum and defined as acting in an unphysical but user-friendly Hilbert space
${\cal R}_N^{(0)}$), the present paper introduces and describes a set of
constructive returns of the description to one of the correct and eligible
physical Hilbert spaces ${\cal R}_0^{(j)}$. The superscript $j$ may run from
$j=0$ to $j=N$. In the $j=0$ extreme of the theory the construction is
currently well known and involves solely the inner product metric
$\Theta=\Theta(H)$. The Hamiltonian $H$ itself remains unchanged. At $j=N$ the
inner-product metric remains trivial and only the Hamiltonian must be
Hermitized, $H \to \mathfrak{h} = \Omega\,H\,\Omega^{-1}=\mathfrak{h}^\dagger$.
At the remaining superscripts $j=1,2,\ldots, N-1$, a new, hybrid form of the
construction of a consistent quantum model is proposed, requiring a
simultaneous amendment of both the metric and the Hamiltonian. In applications,
one of these options is expected to be optimal for a given $H$ in a way
illustrated by a schematic three-state example.
