Abstract
In the popular PT-symmetry-based formulation of quantum mechanics of closed systems one can build unitary models using non-Hermitian Hamiltonians (i.e., H ≠ H^†) which are Hermitizable (so that one can write, simultaneously, H = H). The essence of the trick is that the reference Hilbert space R (in which we use the conventional inner product ψ_a|ψ_b and write H ≠ H^†) is declared unphysical. The necessary Hermiticity of the Hamiltonian H = H can be then achieved by the mere metric-mediated amendment ψ_a|Θ|ψ_b to the inner product. This converts R into a correct physical Hilbert space H. The feasibility of the construction is based on a factorization postulate Θ= PC where, usually, P is parity and C is charge. In our paper we propose a more general factorization recipe in which one constructs Θ=Z_NZ_N-1… Z₁, at any N, in terms of suitable auxiliary pre-metric operators Z_k.