Feasibility and method of multi-step Hermitization of crypto-Hermitian quantum Hamiltonians

In the popular ${\cal PT}-$symmetry-based formulation of quantum mechanics closed systems one can build unitary models using non-Hermitian Hamiltonians (i.e., $H \neq H^\dagger$) which are Hermitizable (so that write, simultaneously, = H^\ddagger$). The essence trick is reference Hilbert space $\cal R$ (in we use conventional inner product $\langle \psi_a|\psi_b\rangle$ and write declared unphysical. necessary Hermiticity Hamiltonian H^\ddagger$ be then achieved by mere metric-mediated amendment \psi_a|\Theta|\psi_b\rangle$ to product. This converts into a correct physical H$. feasibility construction based on factorization postulate $\Theta={\cal PC}$ where, usually, P}$ parity C}$ charge. our paper propose more general recipe in constructs $\Theta=Z_NZ_{N-1}\ldots Z_1$, at any $N$, terms suitable auxiliary pre-metric operators $Z_k$.