An infinitesimal generator of fractional derivatives and integrals

Question

Let $S$ be the unit circle in $\Bbb{R}^2$, and $F$ be the linear space $\{f \in C^{\infty}(S)|\int_{S}f \, d \mu = 0 \}$ where $\mu$ denotes the Hausdorff measure induced by the flat metric of $\Bbb{R}^2$. Let $X=\dfrac{\partial}{\partial \theta}$ be the unit tangent vector field on $S$ and define $T:F \rightarrow F: f \mapsto X(f)$.
Then $F$ becomes a Fréchet space under the total paranorm: $$\lVert f \rVert := \sum_{k=0}^{+ \infty}\dfrac{1}{2^k} \dfrac{\lVert f^{(k)} \rVert_{\infty}}{1+ \lVert f^{(k)} \rVert_{\infty}}$$ where $f^{(k)} := T^{k}(f)$ and $\lVert \cdot \rVert_{\infty}$ denotes the sup-norm. Obviously $\lVert T(f) \rVert \le 2 \lVert f \rVert$ and $T^{-1}(0)= \{ 0 \}$, thus $T$ is a continuous linear automorphism of the Fréchet space $F$.
Question:
Does the definition of fractional derivatives and integrals by Fourier transform on $F$ arise from an infinitesimal generator (i.e. a smooth tangent vector field on $F$)? If not, does there exist a smooth tangent vector field on $F$ such that the one-parameter group generated by it contains $T^{k} \; \forall k \in \Bbb{Z}$?