Factorized Hilbert-space metrics and non-commutative quasi-Hermitian observables

It is well known that an (in general, non-commutative) set of non-Hermitian operators ${\mathrm{\Lambda }}_j$ with real eigenvalues need not necessarily represent observables. We describe a specific class of quantum models in which these operators plus the underlying physical Hilbert-space metric $\mathrm{\Theta }$ are all represented in terms of an auxiliary operator $(N+1)-$plet $Z_k$, $k=0,1,\dots ,N$. Our formalism degenerates to the $\mathcal{P}\mathcal{T}-$symmetric quantum mechanics at $N=2$, with metric $\mathrm{\Theta }=Z_2{Z}_1$, parity $\mathcal{P}=Z_2$, charge $\mathcal{C}=Z_1$ and Hamiltonian $H=Z_0$.