Rigged Hilbert spaces associated with Misra-Prigogine-Courbage theory of irreversibility

摘要

It is proved that, in the Misra-Prigogine-Courbage Theory of Irreversibility using the Internal Time superoperator, fixing its associated non-unitary transformation Lambda, amounts to rigging the corresponding Hilbert-Liouville space, More precisely, it is demonstrated that any Lambda 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 sigma-Hilbertian space, which is a Kothe space if Lambda is compact and is a nuclear space if Lambda has certain nuclear properties: and finally a third one with a smaller sigma-Hilbertian space with a still stronger topology which is nuclear if Lambda " 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 Lambda in a canonical way. (C) 1998 Elsevier Science B.V. All rights reserved. [References: 26]