I revise the question according to the comment of Andrew
Let $D$ be an elliptic differential operator with analytic coefficient on $C^{\infty}(\mathbb{R}^2)$, the space of complex valued smooth functions on $\mathbb{R}^2$.
Let $A\subset C^{\infty}(\mathbb{R}^2) $ be the space of all real analytic $f:\mathbb{R}^2\to \mathbb{C}$ with a global convergence Taylor series. That is $f$ has a global Taylor series representation(with infinite radius of convergence)
Assume that $g\in A,\; f\in C^{\infty}(\mathbb{R}^2)$ with $D(f)=g$ does this imply that $f\in A$?
I am aware of the following regularity property but I am not sure this would answer my question(regarding the infinite radius Taylor expansion):
Regularity: If $g$ is real analytic then $f$ is real analytic.
The motivation for this question is coming from the following post: https://mathoverflow.net/questions/304019/the-comparison-of-certain-modules-arising-from-the-cauchy-riemann-differential-o
Consider the differential operator $$ P = (x^2 + 1)^3(y^2 + 1)\left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} \right)$$ defined on $\mathbb{R}^2$. Notice that this is an elliptic operator with real-analytic coefficient.
Now define an real-analitic function $f$ on $\mathbb{R}^2$ by $$f(x,y) = \frac{1}{(x^2 + 1)} + \frac{1}{(y^2 + 1)}.$$ Notice that Taylor series of $f$ around the origin is finite and if you compute $Pf$ you obtain a real-analytic function whose Taylor series aroung the origin have infinite radius of convergence.
We just constructed a differential operator $P$ and a function $g$ such thatt $g = Pf \in A$ but $f \notin A.$