Let $E,F$ be complex vector bundles over some closed manifold $M$. We investigate operators $T:C^{\infty}(M,E) \to C^{\infty}(M,F)$ between smooth sections of these bundles. We say that such operator is order zero differential operator if $[T,f]=0$ this means that $T$ is a bundle endomorphism. If we know what is order $k$ operator we define $T$ to be order $k+1$ if $[T,f]$ is order $k$. $f$ denotes the smooth function. I would like to understand why this definition implies that every such operator is really a honest differential operator (in local coordinates it has to be ordinary differential operator). I would be grateful if anyone could help me with this.
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2$\begingroup$ A suggestion : If you already now how to prove that any derivation (i.e. such that $\delta(ab)= \delta(a)b+a \delta(b)$ ) on the space of functions is the derivation along a vector fields then you can observe that, (at least when $k=1$) the condition with the commutator is very similar to the property of being a derivation and try the same argument. $\endgroup$– Simon HenryCommented Sep 27, 2016 at 21:31
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1$\begingroup$ You may also find a proof in the book: Nestruev, Smooth Manifolds and Observables, p. 127. springer.com/la/book/9780387955438 $\endgroup$– Michael BächtoldCommented Sep 28, 2016 at 14:55
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First of all, you can show that operators defined by your recursive rule are local operators, i.e., if a section $s$ has support in an open set $U$ then $T(s)$ has support in $U.$ By using the partition of unity it remains to do the work locally. As already mentioned by Simon Henry in his comment, when the order $k=1$, you can directly check that your operator is a first order operator in the sense that it can be written locally as $$(\sum_{i=1}^n A_i\nabla_{X_i})+B$$ for a connection $\nabla$ and suitable homomorphisms $A_i,B\in Hom(E;F).$ This can be generalized by introducing the symbol map for $k$th order differential operators: It can be shown that for functions $f_j,$ $j=1,..k$ the iterated commutator evaluated at $p\in M$ $$[..[T,f_1],..,f_k]$$ is a homomorphism which only depends on $d_pf_1,..,d_pf_k$ in a multilinear symmetric way. Up to a scaling, this commutator is the so called symbol $\sigma$, and it measures the highest order differential operator terms. By subtracting these highest order terms, i.e., something like $$T-constant\sum \sigma(X_{{i_1}}^*,..,X_{{i_k}}^*)\nabla_{X_{i_1}\circ..\circ \nabla_{X_{i_k}}}$$ for a local basis of vector fields and its dual basis, you obtain that the symbol of the difference is $0$, hence it is a differential operator of order $k-1.$
A good reference for this is chapter 2.1. in the book "Heat kernels,.." by Berline, Getzler, Vergne.
