Three weight Koopman semigroups on Lebesgue spaces

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