In this paper, we consider three different semiflows \((\phi _t)_{t\ge 0}, \, (\psi _t)_{t\ge 0}\) and \((\varphi _t)_{t\ge 0}\) on the real half-line given by \(\begin{aligned} \phi _t(r):=e^{-t}r+1-e^{-t},\ \psi _t(r):= \frac{e^t r}{1+r(e^t-1)}, \ \varphi _t(r):= \frac{(1 + e^t)r -1 + e^t}{(-1 + e^t) r + 1 + e^t}, \end{aligned}\) for \(r, t\ge 0\) . These semiflows induce three weight Koopman semigroups, \((T^\gamma _{t, p})_{t>0},\) \( \, (S^\gamma _{t,p})_{t>0}\) and \((R^\gamma _{t,p})_{t>0}\) on the fractional Lebesgue spaces \({\mathcal {T}}_p^{(\alpha )}(t^\alpha )\) , closed subspaces of \(L^p({\mathbb {R}}^+)\) for some \(\alpha \) and \(\gamma \ge 0\) . We describe spectrum sets, point spectrums and resolvent operators of their infinitesimal generators. Three Cesàro-like operators, defined using the Chen fractional integral, \(\begin{aligned} {\mathcal {C}}_{\mu , \nu }f(r):= & \frac{1}{\vert r-1\vert ^{\mu +\nu -1}} \int _{\Gamma _{1,r}}\vert s-1\vert ^{\mu -1} \vert r-s\vert ^{\nu -1} f(s)ds, \quad r>0,\\ {{\mathfrak {C}}}_{\mu , \nu }^\gamma f(r):= & \frac{r^\mu }{\vert r-1\vert ^{\mu +\nu +\gamma -1}} \int _{\Gamma _{1,r}}\frac{\vert s-1\vert ^{\mu +\gamma -1}}{s^{\mu +\nu }} \vert r-s\vert ^{\nu -1} f(s)ds,\ r>0, \\ \textbf{C}^\gamma _{\mu , \nu }f(r):= & \displaystyle 2^\nu \frac{|r + 1|^{\mu - \gamma }}{\ |r-1|^{\mu + \nu - 1}} \int _{\Gamma _{1,r}} \frac{|s - 1|^{\mu - 1}}{\ |s + 1|^{\mu + \nu - \gamma } } \, | r - s|^{\nu - 1} f(s) ds, \ r>0, \\ \end{aligned}\) (for certain \(\mu , \nu , \gamma \in {\mathbb {R}}\) and \(\Gamma _{1,r}:=(1,r)\) when \(r>1\) and \(\Gamma _{1,r}:=(r,1)\) in the case \(0<r<1\) ) are subordinated to these \(C_0\) -semigroups. These representations allow to obtain their norms and spectrum sets.
