Operators which are symmetric but not selfadjoint

Question

Could you please give me some examples of operators defined on inner product space, which are symmetric, but not selfadjoint?

Answer 1

Consider the differential on $L^2(0, \infty)$, i.e. let $\mathcal{D}(A) = C^\infty(0, \infty)$ and $A = i \frac{d}{dx}$. This is symmetric using integration by parts. However, $A$ is not self-adjoint, even more $A$ has no self-adjoint extensions.