Non-Hermitian dynamics and a Hilbert space "relativity principle"

Quantum mechanics (QM) is reexamined from the view-point of operator equivalence classes. Two assumptions are used as starting points:

1.) Any operator with a diagonal spectral decomposition, purely real spectrum and which is densely defined in a suitable Hilbert space may describe a physical observable.
2.) Two sets of observables with operators which are related by a similarity transformation (i.e. which are elements from operator equivalence classes and lie on corresponding conjugacy orbits) are physically indistinguishable. In other words, the physical properties related to the observables X and AXA^-1 are exactly the same when A is a non-singular bounded operator (Hilbert space "relativity principle").
This implies that a given set of observables which is equivalent to a set with all operators Hermitian in a corresponding Hilbert space will not lead to the appearance of new properties compared to conventional QM. In contrast, if a set of observables consists of Hermitian and non-Hermitian operators in a certain Hilbert space then new effects can be expected which will go beyond those of conventional QM.
In this respect we analyze the evolution of a Hermitian observable governed by a non-Hermitian Hamiltonian.
The general approach is illustrated by a toy model 2×2 matrix Hamiltonian as it was recently used by Bender et al [Phys. Rev. Lett. 98 (2007) 040403] for considerations of a PT-symmetric quantum brachistochrone problem. We calculate the corresponding spin flip probabilities and show that the spin flips in non-Hermitian models may have flip (passage) times which are shorter or longer compared to those of Hermitian systems. The effect depends on the concrete exceptional point which is approached in the parameter space of the model.