On spaces $L^p(X)$ the Koopman operator is defined as $T=T_\varphi: L^p(X) \rightarrow L^p(X)$, where $(X,\varphi)$ is a measure preserving system. As $\varphi$ is measure preserving we have that $T$ is an isometry. [1]
On the other hand, let $X$ be a locally compact space and we will work with positive groups on $C_0(X)$. In this context we can define the operator $T(t) f = h_t \cdot (f \circ \phi_t)$, where $\phi_t$ is a flow on $X$ and $(h_t)_{t\in \mathbb{R}} \subset C^b(X)$ the space of continuous bounded functions on $X$. In this case the operator $T(t)$ defines a bounded operator on $C_0(X)$ and if we add some continuity conditions to the flow and cocycle we have that $T(t)$ is a strongly continuous group.[2]
Question
- Is it possible to mix both approaches and consider Koopman operators on $L^p(X)$ defined via a semiflow of measure preserving $\varphi_t$? Or even remove the measure preserving framework and multiply by a nice $h_t$ as we do in the continuous case. Has this been done somewhere?
References
[1] Eisner, T., Farkas, B., Haase, M., & Nagel, R. (2015). Operator theoretic aspects of ergodic theory. (Vol. 272) Springer, Cham.
[2] Arendt, W., Grabosch, A., Greiner, G., Groh, U., Lotz, H., Moustakas, U., Nagel, R., Neubrander, F., & Schlotterbeck, U. (1986). One-parameter semigroups of positive operators. (Vol. 1184) Springer-Verlag, Berlin.
