Some spectral properties of anti-self-adjoint operators on a quaternionic Hibert space

Following the spectral methods established by Adler in his treatment of scattering and decay in the quaternionic quantum theory, it is shown that an anti-self-adjoint operator with quaternion defined spectrum in [0, ∞) has a symmetrical effective spectrum, from the point of view of functional analysis, under quite general conditions. In the case of an operator with absolutely continuous spectrum in [0, ∞), the effective spectrum is absolutely continuous in (- ∞,∞), and a canonically conjugate operator exists. If the anti-self-adjoint operator is the generator of motion in time (Hamiltonian), the conjugate operator is a "time operator." Moreover, according to a theorem of Misra, Prigogine, and Courbage, such a system may admit a Lyapunov operator, and therefore describe irreversible behavior. It is shown directly that no evident contradiction (as is found in the semibounded case in complex quantum mechanics) arises from the definition of a Lyapunov operator in quaternionic quantum mechanics.