If I have an unbounded operator $A$ with domain $D$ on a hilbert space, I can define the sum of $A$ and its adjoint $A^\ast$ on $D$. I know that in general, $A + A^\ast$ will not be self-adjoint, because we only have $(A + B)^\ast \subset A^\ast + B^\ast$.

But for the same reason, $A + A^\ast$ certainly is symmetric. I am wondering if $A + A^\ast$ is also essentially self-adjoint on $D$.

It seems like a simple question but I'm stuck. I also couldn't find anything in books (Reed/Simon, Teschl and whatever I could find on the internet), and the problem is hard to google.
I hope someone can help me out here, or just point me to some book that has information about this topic.

(Additional info: In my specific problem, $A$ is the annihilation operator of the 1D quantum harmonic oscillator (with $D$ taken to be the Schwartz space). But I would be more interested in a general solution ;))