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It is well known that an (in general, non-commutative) set of non-Hermitianoperators $\Lambda_j$ with real eigenvalues need not necessarily representobservables. We describe a specific class of quantum models in which theseoperators plus the underlying physical Hilbert-space metric $\Theta$ are allrepresented in terms of an auxiliary operator $(N+1)-$plet $Z_k$,$k=0,1,\ldots,N$. Our formalism degenerates to the ${\cal PT}-$symmetricquantum mechanics at $N=2$, with metric $\Theta=Z_2Z_1$, parity ${\cal P}=Z_2$,charge ${\cal C}=Z_1$ and Hamiltonian $H=Z_0$.

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