(\[e19a\]), operators $\mathcal{A}$ also evolve according to the conventional Heisenberg-picture equation $$\mathcal{A}(t)=e^{-iHt}\mathcal{A}(0)\,e^{iHt}.
\label{e19b}$$ Given this equation, it is clear how to define an [*observable*]{} in $\mathcal{
PT}$-symmetric quantum mechanics. The crucial property of an observable in any theory of quantum mechanics is that its expectation value in a state must be real. This will be true if $$\mathcal{A}^{\rm T}=\mathcal{A}^{\mathcal{CPT}}=\mathcal{CPT}\mathcal{A}
\,\,\mathcal{CPT},
\label{e19c}$$ where ${\rm T}$ signifies transpose. If this condition is fulfulled by a linear operator in a theory defined by a $\mathcal{PT}$-symmetric Hamiltonian, we say that the operator is “$\mathcal{PCT}$ symmetric” and is an observable. Note that this condition is the analog of the usual condition in conventional quantum mechanics that an observable be Hermitian in the usual Dirac sense $\mathcal{A}^
{\dagger}=\mathcal{A}$: $$\mathcal{A}^{\rm T}=\mathcal{A}^*.
\label{e19d}$$ The condition for an operator to [*remain*]{} an operator as time evolves is simply that $H$ be symmetric: $H^{\rm T}=H$. This symmetry condition has been implicitly assumed in all of the $\mathcal{PT}$-symmetric models discussed in the literature. Recall that the symmetry of $H$ ensures that eigenstates of $H$ corresponding to different energies will be orthogonal. Note that there are two time-independent observables in the theory; namely, $H$ and $\mathcal{C}$.

Illustrative Example: A $2\times2$ Matrix Hamiltonian 