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, Aj≠Aj†, j=1,2,…,K. In principle, the standard probabilistic interpretation of the theory can be re-established via a reconstruction of physical inner-product metric Θ≠I, guaranteeing the quasi-Hermiticity Aj†Θ=ΘAj. The task is easy at K=1 because there are many eligible metrics Θ=Θ(A1). In our paper, the next case with K=2 is analyzed. The criteria of the existence of a shared metric, Θ=Θ(A1,A2), are presented and discussed.