Corollary 3

Let $u$ be a symmetric operator on E. The following conditions are equivalent:

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

(ii) The spaces Ker($u^*+i$) and Ker($u^*-i$) are zero;

(iii) The spaces Im($u+i$) and Im($u-i$) are dense in E;

(iv) The partial symmetric operator $u$ has a unique self-adjoint extension.

Conditions (ii) and (iii) are equivalent by prop. 7, c) of IV, p. 236.

The partial operator $u$ is essentially self-adjoint if and only if $\overline{u}$ is self-adjoint, that is to say if Ker($\overline{u}^*-i$) $=$ Ker($\overline{u}^*+i$) $=\{0\}$ (cor. of prop. 26). Since $\overline{u}^*=u^*$ (prop. 8 of IV, p. 237), condition (i) is therefore equivalent to condition (ii). Moreover, the preceding corollary shows that (ii) implies that $u$ has a unique self-adjoint extension, which is condition (iv).

Finally, suppose that $u$ has a unique self-adjoint extension $v$. Then the same is true of $\overline{u}$, since every self-adjoint extension of $\overline{u}$ is one of $u$, hence is equal to $v$. By the preceding corollary, the spaces Ker($\overline{u}^*+i$) and Ker($\overline{u}^*-i$) must be zero, whence condition (ii), using again $\overline{u}^*=u^*$.