Exercise 13, § 5

Let E be a complex Hilbert space and let $u$ be a symmetric partial operator on E. Suppose that there exists a real number $c >0$ such that $\langle x|u(x)\rangle \geqslant$ $c\|x\|^2$ for all $x\in$ dom($u$).

Now prove that the following conditions are equivalent:

(i) The partial operator $u$ is essentially self-adjoint;

(ii) The image of $u$ is dense in E;

(iii) The kernel of $u^*$ is reduced to 0.