Q:

when is an operator on smooth functions realized by pull-back

I have the following situation: a smooth compact manifold $M$, without boundary and furnished with a smooth volume $\mu $. 
I also have an operator defined on the space of smooth functions with $0$ average w.r.t $\mu$, let's call the space $C^{\infty}_{\mu}(M)$.
This operator is an isomorphism and extends to an isomorphism $H^{s} \rightarrow H^{s}$, for every $s \geq 0$. It also extends to an isometry on $L^{2}$.
My question is the following.  Do there exist conditions under which such an operator is realized by pull-back?  More formally, if an operator $T$ has the above properties (and eventually some additional ones) does there exist a volume preserving diffeomorphism of $M$, $h \in \mathrm{Diff}^{\infty}_{\mu}(M)$ such that for all $\varphi \in C^{\infty}_{\mu}(M)$
\begin{equation}
T.\varphi = \varphi \circ h ^{-1}
\end{equation}
I will probably end up giving an explicit construction of $h$ anyway, but an abstract theorem providing its existence would be helpful. 
Thanks! 

A:

If (and only if) $T$ is also an algebra homomorphism, then it is a pullback by a diffeomorphism.  See the introduction to chapter VIII of here. 
One might be able to replace algebra homomorphism by order respecting properties of $T$.

