Suppose $A$ is densely defined, symmetric ($\langle Ax,y \rangle = \langle x,Ay \rangle$ for $x,y \in \textrm{dom}(A)$), and semi-strictly positive ($\langle Ax, x \rangle > 0$ for all $0 \neq x \in \textrm{dom}(A)$). If $A$ has dense range, is $A$ essentially self-adjoint?
I know that $A$ is essentially self-adjoint if and only if $A + \varepsilon I$ has dense range for all $\varepsilon > 0$. But I was wondering if it is possible for this to fail while $A$ has dense range.