I'd like to define (and calculate the properties of) an object like $e^{\boldsymbol{A\cdot\nabla}}f\left(\boldsymbol{x}\right)$, where $A$ is a collection of matrices written in a vector form, say $\boldsymbol{A\cdot\nabla}=A_{x}\partial_{x}+A_{y}\partial_{y}$ for instance (I think that's the simplest non-trivial example), in which case $f\left(\boldsymbol{x}\right)\equiv f\left(x,y\right)$. The difficulty is that $\left[A_{x},A_{y}\right]\neq0$. $f$ has all the properties one needs (smoothness and co ; it comes from a problem of physics), and it commutes with the $A$'s. The $A$'s do not depend on the coordinates (in the example, this property would read $\partial_{x}A_{x,y}=\partial_{y}A_{x,y}=0$)
Does this object make any sense ? Does it have a name ?
For instance, I'd like to show something like $$e^{\boldsymbol{A\cdot\nabla}}f\left(\boldsymbol{x}\right)e^{-\boldsymbol{A\cdot\nabla}}g\left(\boldsymbol{x}\right)=f\left(\boldsymbol{x}+\boldsymbol{A}\right)g\left(\boldsymbol{x}\right)$$ but this is obviously wrong by direct expansion (it is true when e.g. $A_{y}=0$ in the example above). At second order there are some commutators $\left[A_{i},A_{j}\right]$ appearing.
Any help is warm welcome. (I'm not even sure the tags below are well chosen :-(
Let $l(x,\xi) = \langle x, x^*\rangle + \langle \xi, \xi^*\rangle$ for some fixed $x^*, \xi^* \in \mathbb{R}^d$. Then we get with Weyl quantization that $$(e^{il})^w(x,D) = e^{il(x,D)},$$ where $$e^{il(x,D)}u(x) := e^{i\langle x^*, x\rangle + i/2\langle x^*, \xi^*\rangle}u(x + \xi^*)$$ is the (unique) solution to $i\partial_t v + l(x,D)v = 0$ with initial data $v(0) = u$.
The main point is that you have to use Weyl quantization to get rid of the commutators. The usual pseudodifferential (Kohn-Nirenberg) quantization is not good in that respect.
You can find the above theorem in Zworski - Semiclassical Analysis. Pseudodifferential operators can be found in various books on PDE.
I hope this helps.