Two Koopman semigroups on discrete Lebesgue spaces
Pedro J. Miana
TL;DR
This work builds a bridge between continuous Koopman semigroups on $L^p(\mathbb R^+)$ and discrete $C_0$-semigroups on $\ell^p$ by means of a Poisson transform $\mathcal P$ and its adjoint $\mathcal P^*$. It introduces two concrete Koopman semigroups on $\ell^p$, $\frak T_p(t)$ and $\frak S_p(t)$, that are tightly linked to the classical $T_p(t)$ and $S_p(t)$ on $L^p(\mathbb R^+)$ through explicit intertwining relations with $\mathcal P$ and $\mathcal P^*$. The paper also develops perturbations of these discrete Koopman semigroups and a discrete theory of Cesàro-like operators, including detailed spectral descriptions and connections to Chen fractional integrals. The results provide a coherent discrete-analytic framework for studying dynamical semigroups, enabling transfer of spectral and ergodic properties between $L^p(\mathbb R^+)$ and $\ell^p$ via the Poisson transform. Overall, the work advances operator-theoretic and semigroup techniques in the discrete Lebesgue setting with potential applications to discretized dynamics and approximation theory.
Abstract
In this paper we are interested to connect Koopman semigroups in Lebesgue funcion spaces $L^p(\R^+)$ and $C_0$-semigroups in Lebesgue sequence spaces $\ell^p$ for $1\le p < \infty$. To get this we use certain Poisson transformation $¶: L^p(\mathbb{R}^+)\to \ell^p$ and its adjoint $¶^*$ which allows carry semigroup properties from one space to the other one. Two Koopman semigroups on $\ell^p$ are presented and linked to the standard Koopman semigroup $T_p(t)f(r):= e^{-{t\over p}}f(e^{-t}r)$ and $S_{p}(t)f(r):= e^{-t\over p}f(e^{-t}r+1-e^{-t})$ for $t,r>0$ on $L^p(\R^+)$. In the last section we introduce Cesàro-like operators subordinated to these Koopman semigroups on $\ell^p$.