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.



Abstract definition of differential operators
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.


