Corollary 1

Let $u$ be a closed symmetric operator on a complex Hilbert space E. The mapping $b\mapsto u_b$ induces a bijection of the set of isometries of Ker($u^*-i$) onto Ker($u^*+i$) onto the set of self-adjoint extensions of $u$. In particular, there exists a self-adjoint extension of $u$ if and only if the two components of the deficiency index of $u$ are equal.

This follows from the corollary of proposition 26 and the theorem, together with EVT, V, p. 25, cor. 2.