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

Abstract

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 $Λ$ determinates three canonical riggings of the Liouville space $\QTR{cal}{L}$: a first one with a Hilbert space with a norm greater than the relative one from $\QTR{cal}{L}$; a second one with a $σ$-Hilbertian space, which is a Köthe 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 $Λ^{n}$ is Hilbert-Schmidt, for some positive integer n. Viceversa: any rigging of this type, originated in a dynamical system having an Internal Time superoperator, defines a $Λ$ in a canonical way.