Two Koopman semigroups on discrete Lebesgue spaces

Abstract

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 ${\ell }^p$ for $1\le p<\infty$. To get this we use certain Poisson transformation $\hat{A}\P :L^p({\mathbb{R}}^{+})\to {\ell }^p$ and its adjoint $\hat{A}{\P }^{\ast }$ 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 $T_p(t)f(r):=e^{-\frac{t}{p}}f(e^{-t}r)$ 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 Cesà ro-like operators subordinated to these Koopman semigroups on ${\ell }^p$.