On converse to Koopman's Lemma

✍ Scribed by K. Goodrich; K. Gustafson; B. Misra

• Publisher: Elsevier Science
• Year: 1980
• Tongue: English
• Weight: 636 KB
• Volume: 102
• Category: Article
• ISSN: 0378-4371
• DOI: 10.1016/0378-4371(80)90143-0

✦ Synopsis

Koopman's Lemma states that if a flow T1 is measure preserving for a measure TV on a constant energy surface 0, then the flow generates a one parameter family of unitary operators U, on L* (a, p). We show here a converse, namely that under certain (physically motivated) conditions a unitary operator family U, can be made to generate a corresponding underlying family T, of point transformations.

This result comes out of questions of independent interest in the study of relationships between reversibility and irreversibility, and has application to the foundations of statistical mechanics. In particular, it establishes the principle often used intuitively in chemistry that a forward moving (e.g., Markov) protess that loses information cannot be reversed. In a different setting, it provides the answer to a question in the representation theory of isometries on LP spaces a Banach-Lamperti theorem). These results also allow an interesting reformulation of Ornstein's isomorphism theorem on Bernoulli systems.