On one-parameter Koopman groups

TL;DR

This paper characterizes a specific class of Koopman one-parameter groups on $L_2(X)$, identifying those that preserve $L_(X)$ and have generators as derivations on bounded functions.

Contribution

It provides a complete characterization of unitary Koopman groups that preserve bounded functions and have derivation generators.

Findings

Koopman groups preserving $L_(X)$ are characterized as those with derivation generators.

The paper links the structure of the generator to the preservation of bounded functions.

It advances understanding of the structure of unitary $C_0$-groups in functional analysis.

Abstract

We characterize Koopman one-parameter $C_0$-groups in the class of all unitary one-parameter $C_0$-groups on $L_2(X)$ as those that preserve $L_\infty(X)$ and for which the infinitesimal generator is a derivation on the bounded functions in its domain.