Koopman semigroups in functions and sequences Lebesgue spaces

Composition operators in dynamic systems are often referred to as Koopman operators, in honor of the French-American mathematician Bernard Osgood Koopman (1900–1981). In this talk, we present a brief introduction to Koopman semigroups. Next, we present three examples of weighted Koopman semigroups defined on fractional Lebesgue-Sobolev spaces on the half real line.
We are also interested in connecting Koopman semigroups in Lebesgue spaces of functions $L^p(\mathbb{R}^+)$ and semigroups in Lebesgue spaces of sequences $\ell^p$ for $1\le p < \infty$. To do this, we use a certain Poisson transformation ${\mathcal P}: L^p(\mathbb{R}^+)\to \ell^p$ and its adjoint ${\mathcal P}^*$, which allows us to transfer the properties of the semigroup from one space to another. Two Koopman semigroups in $\ell^p$ are presented and related to the canonical Koopman semigroup in $L^p(\mathbb{R}^+)$.
In the last part of the talk, we introduce operators that extend Cesáro operators (called Chen-type integral operators) subordinate to these Koopman semigroups in $L^p(\mathbb{R}^+)$ and $\ell^p$.
The first results of this article are a joint work with Veronica Poblete, from the University of Chile, and are published in Monatshefte für Mathematik, 206, (2025). The second part of the talk contains results from a preprint available on the Arxiv platform.