Mutual compatibility/incompatibility of quasi-Hermitian quantum observables

In the framework of quasi-Hermitian quantum mechanics the eligible operators of observables may be non-Hermitian, $A_j\mathrm{\ne }{A}_j^{†}$, $j=1,2,\dots ,K$. In principle, the standard probabilistic interpretation of the theory can be re-established via a reconstruction of physical inner-product metric $\mathrm{\Theta }\mathrm{\ne }I$ guaranteeing the quasi-Hermiticity $A_j^{†}\mathrm{\Theta }=\mathrm{\Theta }{A}_j$. The task is easy at $K=1$ because there are many eligible metrics $\mathrm{\Theta }=\mathrm{\Theta }(A_1)$. In our paper the next case with $K=2$ is analyzed. The criteria of the existence of a shared metric $\mathrm{\Theta }=\mathrm{\Theta }(A_1,A_2)$ are presented and discussed.