Two Koopman semigroups on discrete Lebesgue spaces

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 ${\P}: L^p(\mathbb{R}^+)\to \ell^p$ and its adjoint ${\P}^*$ 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 Ces\`aro-like operators subordinated to these Koopman semigroups on $\ell^p$.