Two Koopman semigroups on discrete Lebesgue spaces

In this paper we are interested to connect Koopman semigroups in Lebesgue funcion spaces $L^p({\mathbb{R}}^{+})$ and $C_0$-semigroups in Lebesgue sequence spaces ${\mathrm{\ell }}^p$ for $1\le p<\mathrm{\infty }$. To get this we use certain Poisson transformation $\mathrm{\P }:L^p({\mathbb{R}}^{+})\to {\mathrm{\ell }}^p$ and its adjoint ${\mathrm{\P }}^{\ast }$ which allows carry semigroup properties from one space to the other one. Two Koopman semigroups on ${\mathrm{\ell }}^p$ are presented and linked to the standard Koopman semigroup $T_p(t)f(r):=e^{-\frac{t}{p}}f(e^{-t}r)$ and $S_p(t)f(r):=e^{\frac{-t}{p}}f(e^{-t}r+1-e^{-t})$ for $t,r>0$ on $L^p({\mathbb{R}}^{+})$. In the last section we introduce Cesàro-like operators subordinated to these Koopman semigroups on ${\mathrm{\ell }}^p$.