Rigged Hilbert Spaces associated with Misra-Prigogine-Courbage Theory of Irreversibility

It is proved that, in the Misra–Prigogine–Courbage Theory of Irreversibility using the Internal Time superoperator, fixing its associated non-unitary transformation Λ, amounts to rigging the corresponding Hilbert–Liouville space. More precisely, it is demonstrated that any Λ determines three canonical riggings of the Liouville space L: first one with a Hilbert space with a norm greater than the relative one from L; a second one with a σ-Hilbertian space, which is a Kothe space if Λ is compact and is a nuclear space if Λ has certain nuclear properties; and finally a third one with a smaller σ-Hilbertian space with a still stronger topology which is nuclear if Λn is Hilbert–Schmidt, for some positive integer n. In contrast any rigging of this type, originated in a dynamical system having an Internal Time superoperator, defines a Λ in a canonical way.