Definition 10

Let $u$ be a closed symmetric operator on a complex Hilbert space E. Let $E_+=$ Ker($u^*-i$) and $E_-=$ Ker($u^*+i$). The pair $(E_+,E_-)$ of subspaces of dom($u^*$) is called the deficiency pair of $u$. The subspaces $E_+$ and $E_-$ are closed subspaces of E (prop. 3 of IV, p. 229). The pair (dim(E$_+$), dim(E$_-$)) of the Hilbert dimensions of these subspaces is called the deficiency index of $u$.