# Mutual Compatibility/Incompatibility of Quasi-Hermitian Quantum Observables
## 1 Introduction  
It used to be widely accepted that a consistent picture of a unitary quantum system can only be based on our knowledge of its observable quantities represented by a set of operators \(A_{1}\), \(A_{2}\ldots\) that are all self-adjoint in a suitable Hilbert space (cf., e.g., [1]). More than thirty years ago, [PERSON], [PERSON], and [PERSON] [2] pointed out that such a statement may lead to misunderstandings because, in some fairly realistic models, the operators of observables could be defined as acting, simultaneously, in _two different_ Hilbert spaces, viz., in spaces \(\mathcal{H}_{mathematical}\) and \(\mathcal{H}_{physical}\).  
By construction, the latter two spaces have to coincide as the sets of ket vectors \(|\psi\rangle\). Still, they may be unitarily non-equivalent because they may differ by the use of the two alternative forms of the respective inner products. Thus, in order to avoid misunderstandings, one has to speak, first, about a maximally user-friendly inner product that specifies a manifestly unphysical Hilbert space \(\mathcal{H}_{mathematical}\). At the same time, one also has to speak about the second correct physical inner product that must be used to specify the other presumably user-unfriendly Hilbert space of state \(\mathcal{H}_{physical}\), which is still the only one that provides their correct probabilistic interpretation.  
The technical details of the idea may be found in [2]. A decisive advantage of keeping the two spaces different has been found in the possibility of reducing the difference to the mere redefinition of the inner product,  
\[\langle\psi_{a}|\psi_{b}\rangle_{mathematical}\,\rightarrow\,\langle\psi_{a}| \psi_{b}\rangle_{physical}\,\equiv\,\langle\psi_{a}|\Theta|\psi_{b}\rangle_{mathematical}\,. \tag{1}\]  
The amended (often called physical) inner-product metric \(\Theta\
eq I\) should be a positive-definite and bounded operator that is self-adjoint in \(\mathcal{H}_{mathematical}\) and has a boundedinverse [3]. Under these assumptions, the \"obligatory\" requirement of Hermiticity of \(A_{j}\)s in \(\mathcal{H}_{physical}\) can be re-expressed, in the preferred representation space \(\mathcal{H}_{mathematical}\), as the quasi-Hermiticity [4]_alias_\(\Theta-\)pseudo-Hermiticity [5],  
\[A_{j}^{\dagger}\,\Theta=\Theta\,A_{j}\,,\quad\forall j\,. \tag{2}\]  
Needless to add that the two Hilbert spaces would coincide in the limit of \(\Theta\to I\). Within the generic theory with \(\Theta\
eq I\), nevertheless, the \"hidden Hermiticity\" constraint (2) will still imply the reality of the spectrum.  
In most applications, the model-building process proceeds in the opposite direction. In the manner sampled in [2] or [6], one is just given a set \(\{A_{j}\}\) of the candidates for observables. All of these operators have to be treated, in \(\mathcal{H}_{mathematical}\), as non-Hermitian, \(A_{j}\
eq A_{j}^{\dagger}\). For all of them, the reality of the spectrum is a merely necessary, but not sufficient, condition of their observability and mutual compatibility. One has to prove the mathematical consistency of the theory by showing that _all_ of the preselected operators of observables _share_ the quasi-Hermiticity property (2). A clarification of these items is also the main purpose of our present paper.  
In our preceding study [7] of the problem, we showed that, even when one decides to work with the mere doublet of the \"input\" operators \(A_{1}\) and \(A_{2}\), the acceptable inner-product metric need not exist at all. The essence of such a no-go theorem was that the two \"arbitrary\" operators, \(A_{1}\) and \(A_{2}\), had to be quasi-Hermitian with respect to a single subscript-independent metric \(\Theta\). This simply imposes too many constraints upon \(\Theta\) in general.  
In what follows, we will outline a continuation of the latter study. We will introduce and describe an efficient strategy for the construction of \(\Theta\) whenever it does exist.