Self adjoint, positive definite, non elliptic operator on Hilbert space

Question

I'm trying to find an example of a Hilbert space and a self adjoint, positive definite operator $A\in L(H)$ that is not elliptic. A suggestion given is to look at the space of square summable sequences $\ell^{2}$ and a linear operator that dilates the $n$th standard basis vector $e_{n}=\mathbb{I}[n=i]$ by a small positive factor. I still cannot see how to achieve this, so any help would be appreciated.